C1.6 Transform polynomial function graphs | Ontario Grade 12 Mathematics

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

How changes to a polynomial rule move, stretch, and reflect its graph

A polynomial graph can be changed by adjusting its function rule. The changes may move the graph, make it wider or narrower, or reflect it across an axis. These transformations apply to polynomials of different degrees. The same point-mapping ideas work for a quadratic, cubic, or other polynomial. In this lesson, you will connect the rule, a table of points, and the resulting graph.

What you will learn

Identify translations, stretches, and reflections in a transformed polynomial function.
Describe how a point on an original graph maps to a point on a transformed graph.
Use a table of points to sketch and check a transformed polynomial graph.

1. Prerequisite bridge: functions and points

A function rule assigns an output to each allowed input. For example, if

f(x)=x^3-3x

, then

f(2)=2

. A point on its graph has the form

(x,f(x))

, so

(2,2)

lies on the graph.

A transformation changes the location or shape of a graph. A translation moves every point the same distance. A reflection flips the graph across a line, such as the

x

-axis. A stretch changes distances from an axis; a compression makes those distances smaller.

The graph being changed is the original graph,

y=f(x)

. The new graph is described using a transformed rule. Comparing a few corresponding points is a practical way to see what the rule does.

(x,y)=(x,f(x))

The first coordinate of a graph point is the input; the second is the output.
A transformation rule changes the coordinates of points from the original graph.

2. Read the transformations in the rule

A useful general form is

g(x)=a f(k(x-d))+c

. Here,

f

is the original polynomial rule and

g

is the transformed rule. The value

d

shifts the graph horizontally, and

c

shifts it vertically. A positive

d

moves it right; a negative

d

moves it left. A positive

c

moves it up; a negative

c

moves it down.

The factor

a

changes the outputs. If

|a|>1

, the graph is stretched vertically by that factor. If

0<|a|<1

, it is compressed vertically. If

a<0

, it is also reflected across the

x

-axis.

The factor

k

changes the inputs inside the function. Its effect on horizontal distances is the reciprocal: the horizontal scale factor is

1/|k|

. If

|k|>1

, the graph is compressed horizontally. If

0<|k|<1

, it is stretched horizontally. When

k<0

, the graph is also reflected horizontally; the point mapping captures this direction change.

For a point

(x,y)

on

y=f(x)

, the corresponding point on

y=g(x)

is

(d+x/k, ay+c)

. This follows by matching the input to

f

: the new input must make

k(x_{ ext{new}}-d)

equal to the original input

x

. Then the output is multiplied by

a

and shifted by

c

.

Apply the transformations to the graph as a whole. When describing a combined transformation, remember that the input changes inside

f

and the output changes outside it. A table helps prevent errors with horizontal scaling.

(x,y)\mapsto\left(d+\frac{x}{k},\ ay+c\right)

Outside $f$ , $a$ changes vertical scale or reflects; $c$ shifts vertically.
Inside $f$ , $k$ changes horizontal scale or reflects; $d$ shifts horizontally.
The horizontal scale factor is reciprocal to the size of $k$ .

3. Guided example: combine changes and map points

Start with the polynomial

f(x)=x^3-3x

. The new rule is

g(x)=-2f\left(\frac{x-1}{2}\right)+3

. Match it to the general form:

a=-2

,

k=\frac12

,

d=1

, and

c=3

.

The graph is stretched horizontally by a factor of

2

, shifted right

1

, stretched vertically by a factor of

2

, reflected across the

x

-axis, and shifted up

3

. The horizontal stretch comes from the reciprocal of

\frac12

.

Choose input values for

f

and calculate their outputs. Then map each original point to the new graph. The table shows that the horizontal coordinate changes to

1+2x

, while the output changes to

-2y+3

.

The listed points are useful for sketching, but they do not by themselves specify every point on the curve. Draw a smooth cubic shape through them, preserving the graph's overall pattern after the stated transformations.

g(x)=-2f\left(\frac{x-1}{2}\right)+3

Work from points on the original polynomial.
Use the point-mapping rule for both coordinates.
Use the rule's signs to describe reflections and translations.

4. Apply the idea and check your graph

To sketch another transformed polynomial, identify the original rule first. Read the numbers and signs outside and inside the function. Describe the vertical changes and horizontal changes, then use a few points to locate the new graph.

A point can also help you check whether a proposed sketch is consistent. If an original point has output zero, its transformed output is

c

, because multiplying zero by

a

still gives zero. Its horizontal position may still change. This is a quick check on the vertical translation, not a claim that all transformed graphs have a point at that height.

If a polynomial is given expanded rather than in the form

a f(k(x-d))+c

, identify the original function or rewrite the rule before describing transformations. Do not assume that a coefficient multiplying the whole polynomial is a horizontal change; it changes outputs.

g(x)=a f(k(x-d))+c

A table can test whether a sketch agrees with the transformed rule.
The form of the rule helps distinguish input changes from output changes.

Original points and their transformed positions

Original input $x$	Original output $f(x)$	Mapped input $1+2x$	Mapped output $-2f(x)+3$
$-2$	$-2$	$-3$	$7$
$-1$	$2$	$-1$	$-1$
$0$	$0$	$1$	$3$
$1$	$-2$	$3$	$7$
$2$	$2$	$5$	$-1$

Worked example

Map a cubic graph through combined transformations

For

f(x)=x^3-3x

and

g(x)=-2f\left(\frac{x-1}{2}\right)+3

, describe the transformations and map the original points with inputs

-2,-1,0,1,2

.

Identify the transformation values
Compare the rule with $g(x)=a f(k(x-d))+c$ . This gives $a=-2$ , $k=\frac12$ , $d=1$ , and $c=3$ . The negative $a$ reflects across the $x$ -axis as well as stretching vertically.
$a=-2,\quad k=\frac12,\quad d=1,\quad c=3$
Describe the graph changes
The horizontal scale factor is $1/|k|=2$ , so the graph is stretched horizontally by $2$ . It shifts right $1$ . The vertical factor has magnitude $2$ , so it stretches vertically by $2$ , reflects across the $x$ -axis, and then shifts up $3$ .
$\frac{1}{|k|}=2$
Find original outputs
Substitute each chosen input into $f(x)=x^3-3x$ . For example, $f(-2)=-8+6=-2$ , and $f(2)=8-6=2$ . The other outputs are $2,0,-2$ for inputs $-1,0,1$ .
$f(-2)=-2,\quad f(-1)=2,\quad f(0)=0,\quad f(1)=-2,\quad f(2)=2$
Map each point
For each original point $(x,y)$ , use $(1+2x,-2y+3)$ . This gives the new horizontal coordinate from the horizontal stretch and shift, and the new output from the reflection, vertical stretch, and upward shift.
$(x,y)\mapsto(1+2x,-2y+3)$

Answer: The graph is stretched horizontally by

2

, shifted right

1

, stretched vertically by

2

, reflected across the

x

-axis, and shifted up

3

. The mapped points are

(-3,7)

,

(-1,-1)

,

(1,3)

,

(3,7)

, and

(5,-1)

, in the order of the original inputs

-2,-1,0,1,2

.

Check: For the original point

(0,0)

, the map gives

(1,3)

. The transformed rule also gives

g(1)=-2f(0)+3=3

, so this mapped point checks.

Common mistakes and how to avoid them

Treating the horizontal factor

k

as the horizontal scale factor.

Correction: The horizontal scale factor is

1/|k|

. For example,

k=\frac12

gives a horizontal stretch by

2

.

Thinking that a negative factor inside

f

reflects across the

x

-axis.

Correction: A negative factor inside the function reflects horizontally. A negative outside factor,

a

, reflects across the

x

-axis.

Applying the vertical shift before multiplying the output.

Correction: Use the mapped output

ay+c

: first multiply the original output by

a

, then add

c

.

Moving points left when

d

is positive.

Correction: In

f(k(x-d))

, positive

d

shifts the graph right by

d

.

Lesson summary

The form $g(x)=a f(k(x-d))+c$ displays common transformations of a polynomial graph.
The values $d$ and $c$ shift the graph horizontally and vertically.
The value $a$ changes vertical scale and may reflect across the $x$ -axis.
The value $k$ changes horizontal scale and may reflect horizontally; the scale factor is $1/|k|$ .
Map points with $(x,y)\mapsto(d+x/k,ay+c)$ , then use them to sketch and check the transformed graph.

Check your understanding

Question 1

For

g(x)=3f(x+2)-1

, which description is correct?

Shift left $2$ , stretch vertically by $3$ , and shift down $1$ .
Shift right $2$ , stretch horizontally by $3$ , and shift down $1$ .
Shift left $2$ , reflect across the $x$ -axis, and shift up $1$ .
correctIndex

Show answer and explanation

Shift left

2

, stretch vertically by

3

, and shift down

1

.

Since

x+2=x-(-2)

, the graph shifts left

2

. The outside factor

3

stretches vertically by

3

, and

-1

shifts it down

1

.

Question 2

In

g(x)=f(2(x-1))

, what is the horizontal scale factor?

$2$ , so the graph stretches horizontally.
$\frac12$ , so the graph compresses horizontally.
$2$ , so the graph compresses vertically.
correctIndex

Show answer and explanation

\frac12

, so the graph compresses horizontally.

Here

k=2

, so the horizontal scale factor is

1/|2|=\frac12

. The graph is compressed horizontally.

Question 3

The point

(4,-1)

is on

y=f(x)

. Under

g(x)=-f(x-3)+2

, where does it map?

$(7,3)$
$(1,3)$
$(7,1)$
correctIndex

Show answer and explanation

(7,3)

Here

a=-1

,

k=1

,

d=3

, and

c=2

. The point maps to

(3+4,-(-1)+2)=(7,3)

.

Key terms

Polynomial function: A function made from terms with real-number coefficients and non-negative whole-number powers of the variable.
Translation: A move of a graph that shifts every point the same distance in the same direction.
Reflection: A flip of a graph across a line, such as the $x$ -axis.
Vertical stretch or compression: A change that multiplies the graph's output distances from the $x$ -axis by a factor.
Horizontal stretch or compression: A change that multiplies the graph's input distances from the vertical line of reflection or shift by a factor.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C1.6. It is a study resource, not an official curriculum publication.

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C1.6 · Transform polynomial function graphs

Ontario Grade 12 Mathematics

How changes to a polynomial rule move, stretch, and reflect its graph

What you will learn

1. Prerequisite bridge: functions and points

2. Read the transformations in the rule

3. Guided example: combine changes and map points

4. Apply the idea and check your graph

Original points and their transformed positions

Map a cubic graph through combined transformations

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review