Practice · Trade a vertical shift for a horizontal one · Sensim Math

Variant 1 easy

The graph of the linear function $y=3x+1$ is translated $2$ units in the direction of the $x$ -axis and $-1$ unit in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=3x+1$ right $2$ and down $1$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=3x+1$ .
First move: $2$ units along the $x$ -axis and $-1$ unit along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $3$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $3$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

2

along the

x

-axis replaces

x

with

x-2

, and moving

-1

along the

y

-axis subtracts

1

from the output.

y=3(x-2)+1-1=3x-6

The new line has slope

3

and

y

-intercept

-6

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=3(x-m)+1=3x-3m+1

Same slope

3

; the

y

-intercept is now

-3m+1

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-3m+1=-6 \Rightarrow 3m=7 \Rightarrow m=\frac{7}{3}

One linear equation in

m

finishes it.

Answer:

m=\frac{7}{3}

4 · Reviewdoes it hold up?

Check: $3\left(x-\frac{7}{3}\right)+1=3x-7+1=3x-6$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $3$ , dropping $1$ unit is the same as sliding $\frac{1}{3}$ unit to the right. So the down- $1$ move is worth an extra $\frac{1}{3}$ to the right, and $m=2+\frac{1}{3}=\frac{7}{3}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 2 easy

The graph of the linear function $y=2x-1$ is translated $3$ units in the direction of the $x$ -axis and $3$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=2x-1$ right $3$ and up $3$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=2x-1$ .
First move: $3$ units along the $x$ -axis and $3$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $2$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $2$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

3

along the

x

-axis replaces

x

with

x-3

, and moving

3

along the

y

-axis adds

3

to the output.

y=2(x-3)-1+3=2x-4

The new line has slope

2

and

y

-intercept

-4

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=2(x-m)-1=2x-2m-1

Same slope

2

; the

y

-intercept is now

-2m-1

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-2m-1=-4 \Rightarrow 2m=3 \Rightarrow m=\frac{3}{2}

One linear equation in

m

finishes it.

Answer:

m=\frac{3}{2}

4 · Reviewdoes it hold up?

Check: $2\left(x-\frac{3}{2}\right)-1=2x-3-1=2x-4$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $2$ , rising $3$ units is the same as sliding $\frac{3}{2}$ units to the left. So the up- $3$ move takes back $\frac{3}{2}$ of the move to the right, and $m=3-\frac{3}{2}=\frac{3}{2}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 3 easy

The graph of the linear function $y=2x+3$ is translated $1$ unit in the direction of the $x$ -axis and $-1$ unit in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=2x+3$ right $1$ and down $1$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=2x+3$ .
First move: $1$ unit along the $x$ -axis and $-1$ unit along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $2$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $2$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

1

along the

x

-axis replaces

x

with

x-1

, and moving

-1

along the

y

-axis subtracts

1

from the output.

y=2(x-1)+3-1=2x

The new line has slope

2

and

y

-intercept

0

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=2(x-m)+3=2x-2m+3

Same slope

2

; the

y

-intercept is now

-2m+3

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-2m+3=0 \Rightarrow 2m=3 \Rightarrow m=\frac{3}{2}

One linear equation in

m

finishes it.

Answer:

m=\frac{3}{2}

4 · Reviewdoes it hold up?

Check: $2\left(x-\frac{3}{2}\right)+3=2x-3+3=2x$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $2$ , dropping $1$ unit is the same as sliding $\frac{1}{2}$ unit to the right. So the down- $1$ move is worth an extra $\frac{1}{2}$ to the right, and $m=1+\frac{1}{2}=\frac{3}{2}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 4 medium

The graph of the linear function $y=3x-4$ is translated $4$ units in the direction of the $x$ -axis and $-2$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=3x-4$ right $4$ and down $2$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=3x-4$ .
First move: $4$ units along the $x$ -axis and $-2$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $3$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $3$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

4

along the

x

-axis replaces

x

with

x-4

, and moving

-2

along the

y

-axis subtracts

2

from the output.

y=3(x-4)-4-2=3x-18

The new line has slope

3

and

y

-intercept

-18

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=3(x-m)-4=3x-3m-4

Same slope

3

; the

y

-intercept is now

-3m-4

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-3m-4=-18 \Rightarrow 3m=14 \Rightarrow m=\frac{14}{3}

One linear equation in

m

finishes it.

Answer:

m=\frac{14}{3}

4 · Reviewdoes it hold up?

Check: $3\left(x-\frac{14}{3}\right)-4=3x-14-4=3x-18$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $3$ , dropping $2$ units is the same as sliding $\frac{2}{3}$ unit to the right. So the down- $2$ move is worth an extra $\frac{2}{3}$ to the right, and $m=4+\frac{2}{3}=\frac{14}{3}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 5 easy

The graph of the linear function $y=4x-2$ is translated $3$ units in the direction of the $x$ -axis and $1$ unit in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=4x-2$ right $3$ and up $1$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=4x-2$ .
First move: $3$ units along the $x$ -axis and $1$ unit along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $4$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $4$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

3

along the

x

-axis replaces

x

with

x-3

, and moving

1

along the

y

-axis adds

1

to the output.

y=4(x-3)-2+1=4x-13

The new line has slope

4

and

y

-intercept

-13

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=4(x-m)-2=4x-4m-2

Same slope

4

; the

y

-intercept is now

-4m-2

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-4m-2=-13 \Rightarrow 4m=11 \Rightarrow m=\frac{11}{4}

One linear equation in

m

finishes it.

Answer:

m=\frac{11}{4}

4 · Reviewdoes it hold up?

Check: $4\left(x-\frac{11}{4}\right)-2=4x-11-2=4x-13$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $4$ , rising $1$ unit is the same as sliding $\frac{1}{4}$ unit to the left. So the up- $1$ move takes back $\frac{1}{4}$ of the move to the right, and $m=3-\frac{1}{4}=\frac{11}{4}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 6 medium

The graph of the linear function $y=4x+3$ is translated $1$ unit in the direction of the $x$ -axis and $-3$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=4x+3$ right $1$ and down $3$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=4x+3$ .
First move: $1$ unit along the $x$ -axis and $-3$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $4$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $4$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

1

along the

x

-axis replaces

x

with

x-1

, and moving

-3

along the

y

-axis subtracts

3

from the output.

y=4(x-1)+3-3=4x-4

The new line has slope

4

and

y

-intercept

-4

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=4(x-m)+3=4x-4m+3

Same slope

4

; the

y

-intercept is now

-4m+3

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-4m+3=-4 \Rightarrow 4m=7 \Rightarrow m=\frac{7}{4}

One linear equation in

m

finishes it.

Answer:

m=\frac{7}{4}

4 · Reviewdoes it hold up?

Check: $4\left(x-\frac{7}{4}\right)+3=4x-7+3=4x-4$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $4$ , dropping $3$ units is the same as sliding $\frac{3}{4}$ unit to the right. So the down- $3$ move is worth an extra $\frac{3}{4}$ to the right, and $m=1+\frac{3}{4}=\frac{7}{4}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 7 hard

The graph of the linear function $y=5x-3$ is translated $1$ unit in the direction of the $x$ -axis and $2$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=5x-3$ right $1$ and up $2$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=5x-3$ .
First move: $1$ unit along the $x$ -axis and $2$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $5$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $5$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

1

along the

x

-axis replaces

x

with

x-1

, and moving

2

along the

y

-axis adds

2

to the output.

y=5(x-1)-3+2=5x-6

The new line has slope

5

and

y

-intercept

-6

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=5(x-m)-3=5x-5m-3

Same slope

5

; the

y

-intercept is now

-5m-3

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-5m-3=-6 \Rightarrow 5m=3 \Rightarrow m=\frac{3}{5}

One linear equation in

m

finishes it.

Answer:

m=\frac{3}{5}

4 · Reviewdoes it hold up?

Check: $5\left(x-\frac{3}{5}\right)-3=5x-3-3=5x-6$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $5$ , rising $2$ units is the same as sliding $\frac{2}{5}$ unit to the left. So the up- $2$ move takes back $\frac{2}{5}$ of the move to the right, and $m=1-\frac{2}{5}=\frac{3}{5}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 8 medium

The graph of the linear function $y=5x+2$ is translated $2$ units in the direction of the $x$ -axis and $-3$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=5x+2$ right $2$ and down $3$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=5x+2$ .
First move: $2$ units along the $x$ -axis and $-3$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $5$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $5$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

2

along the

x

-axis replaces

x

with

x-2

, and moving

-3

along the

y

-axis subtracts

3

from the output.

y=5(x-2)+2-3=5x-11

The new line has slope

5

and

y

-intercept

-11

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=5(x-m)+2=5x-5m+2

Same slope

5

; the

y

-intercept is now

-5m+2

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-5m+2=-11 \Rightarrow 5m=13 \Rightarrow m=\frac{13}{5}

One linear equation in

m

finishes it.

Answer:

m=\frac{13}{5}

4 · Reviewdoes it hold up?

Check: $5\left(x-\frac{13}{5}\right)+2=5x-13+2=5x-11$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $5$ , dropping $3$ units is the same as sliding $\frac{3}{5}$ unit to the right. So the down- $3$ move is worth an extra $\frac{3}{5}$ to the right, and $m=2+\frac{3}{5}=\frac{13}{5}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 9 medium

The graph of the linear function $y=3x+5$ is translated $1$ unit in the direction of the $x$ -axis and $2$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=3x+5$ right $1$ and up $2$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=3x+5$ .
First move: $1$ unit along the $x$ -axis and $2$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $3$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $3$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

1

along the

x

-axis replaces

x

with

x-1

, and moving

2

along the

y

-axis adds

2

to the output.

y=3(x-1)+5+2=3x+4

The new line has slope

3

and

y

-intercept

4

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=3(x-m)+5=3x-3m+5

Same slope

3

; the

y

-intercept is now

-3m+5

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-3m+5=4 \Rightarrow 3m=1 \Rightarrow m=\frac{1}{3}

One linear equation in

m

finishes it.

Answer:

m=\frac{1}{3}

4 · Reviewdoes it hold up?

Check: $3\left(x-\frac{1}{3}\right)+5=3x-1+5=3x+4$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $3$ , rising $2$ units is the same as sliding $\frac{2}{3}$ unit to the left. So the up- $2$ move takes back $\frac{2}{3}$ of the move to the right, and $m=1-\frac{2}{3}=\frac{1}{3}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 10 hard

The graph of the linear function $y=6x-5$ is translated $3$ units in the direction of the $x$ -axis and $5$ units in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=6x-5$ right $3$ and up $5$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=6x-5$ .
First move: $3$ units along the $x$ -axis and $5$ units along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $6$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $6$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

3

along the

x

-axis replaces

x

with

x-3

, and moving

5

along the

y

-axis adds

5

to the output.

y=6(x-3)-5+5=6x-18

The new line has slope

6

and

y

-intercept

-18

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=6(x-m)-5=6x-6m-5

Same slope

6

; the

y

-intercept is now

-6m-5

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-6m-5=-18 \Rightarrow 6m=13 \Rightarrow m=\frac{13}{6}

One linear equation in

m

finishes it.

Answer:

m=\frac{13}{6}

4 · Reviewdoes it hold up?

Check: $6\left(x-\frac{13}{6}\right)-5=6x-13-5=6x-18$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $6$ , rising $5$ units is the same as sliding $\frac{5}{6}$ unit to the left. So the up- $5$ move takes back $\frac{5}{6}$ of the move to the right, and $m=3-\frac{5}{6}=\frac{13}{6}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 11 hard

The graph of the linear function $y=6x+1$ is translated $2$ units in the direction of the $x$ -axis and $-1$ unit in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=6x+1$ right $2$ and down $1$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=6x+1$ .
First move: $2$ units along the $x$ -axis and $-1$ unit along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $6$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $6$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

2

along the

x

-axis replaces

x

with

x-2

, and moving

-1

along the

y

-axis subtracts

1

from the output.

y=6(x-2)+1-1=6x-12

The new line has slope

6

and

y

-intercept

-12

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=6(x-m)+1=6x-6m+1

Same slope

6

; the

y

-intercept is now

-6m+1

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-6m+1=-12 \Rightarrow 6m=13 \Rightarrow m=\frac{13}{6}

One linear equation in

m

finishes it.

Answer:

m=\frac{13}{6}

4 · Reviewdoes it hold up?

Check: $6\left(x-\frac{13}{6}\right)+1=6x-13+1=6x-12$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $6$ , dropping $1$ unit is the same as sliding $\frac{1}{6}$ unit to the right. So the down- $1$ move is worth an extra $\frac{1}{6}$ to the right, and $m=2+\frac{1}{6}=\frac{13}{6}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Variant 12 hard

The graph of the linear function $y=4x+7$ is translated $2$ units in the direction of the $x$ -axis and $-1$ unit in the direction of the $y$ -axis. The resulting graph is exactly the same as the graph you get by translating the original graph $m$ units in the direction of the $x$ -axis only. Find the value of $m$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Shift the line $y=4x+7$ right $2$ and down $1$ . That same final line can also be reached by a single sideways shift of $m$ units. Find $m$ .

Givens

The original line is $y=4x+7$ .
First move: $2$ units along the $x$ -axis and $-1$ unit along the $y$ -axis.
Second move: $m$ units along the $x$ -axis only, landing on the same line.

Unknowns

The horizontal shift $m$ .

Constraints

Translating a line never changes its slope, so both results have slope $4$ ; they match exactly when their $y$ -intercepts match.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#15 Organize Information in More Ways

Write each translated line as an equation (shifting by $p$ along $x$ replaces $x$ with $x-p$ ; shifting by $q$ along $y$ adds $q$ ). Both lines keep slope $4$ , so setting their $y$ -intercepts equal gives one equation for $m$ .

3 · Execute3 carry out the plan

1Translate the first way

#13 Convert to Algebra 8.F.A.3

Moving

2

along the

x

-axis replaces

x

with

x-2

, and moving

-1

along the

y

-axis subtracts

1

from the output.

y=4(x-2)+7-1=4x-2

The new line has slope

4

and

y

-intercept

-2

.

2Translate the second way

#13 Convert to Algebra 8.F.A.3

Moving

m

along the

x

-axis only replaces

x

with

x-m

.

y=4(x-m)+7=4x-4m+7

Same slope

4

; the

y

-intercept is now

-4m+7

.

3Match the intercepts

#15 Organize Information in More Ways 8.F.B.4

Two lines with the same slope coincide exactly when their

y

-intercepts are equal.

-4m+7=-2 \Rightarrow 4m=9 \Rightarrow m=\frac{9}{4}

One linear equation in

m

finishes it.

Answer:

m=\frac{9}{4}

4 · Reviewdoes it hold up?

Check: $4\left(x-\frac{9}{4}\right)+7=4x-9+7=4x-2$ , the same line as the first translation.

Another way: Picture it (Draw a Diagram): on a line of slope $4$ , dropping $1$ unit is the same as sliding $\frac{1}{4}$ unit to the right. So the down- $1$ move is worth an extra $\frac{1}{4}$ to the right, and $m=2+\frac{1}{4}=\frac{9}{4}$ .

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Writing each translated line in the form $y=mx+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Reading off and comparing the $y$-intercepts of the two lines.

💡Takeaway. On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.

Trade a vertical shift for a horizontal one · 12 practice problems

Generated variants — 12

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