Polynomial functions

Chapter 5 · 6 lessons · 204 questions

Unit 1: Function notation, tables and graphs

Key points

$f(x)$ is the output of the function $f$ when the input is $x$ .
To find $f(3)$ , replace every $x$ in the rule with $3$ .
An expression goes in whole: for $f(x + 1)$ , replace every $x$ with $(x + 1)$ .
$f(k) = m$ is the point $(k, m)$ on the graph of $f$ .

See it solved

Example

The function $f$ is defined by $f(x)$ $= 2x^2 - 3x + 1$ . What is the value of $f(-2)$ ?

Worked by hand

The plan

Plug the input into the function

Desmos: Desmos is quicker and avoids sign slips: define $f$ once, then type $f(-2)$ .

Step 1: Plug the input into the function
- Replace every $x$ with $(-2)$ , one term at a time.
- First term, $2x^2$ : $(-2)^2 = 4$ , so $2 \cdot 4 = 8$ .
- Second term, $-3x$ : $-3 \cdot (-2) = 6$ .
- The last term is $1$ .
- $f(-2)$ $= 8 + 6 + 1$ $= 15$ .

The correct answer is

15

The same example in Desmos

new graph
Keys: f(x)=2x^2 → -3x+1
Desmos shows: The function, as a parabola.
Desmos shows: $15$ , at the right of the row.

Desmos expression list: f(x)=2x^2-3x+1; f(-2) — Type

Desmos graph of f(x)=2x^2-3x+1; f(-2), with (−2, 15) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $g$ is defined by $g(x) = 3x - 7$ . For what value of $x$ does $g(x) = 11$ ?

Show the answerHide the answer

Answer

The correct answer is $6$ .

Step 1: Set the function equal to the output
- $g(x) = 11$ gives the output. The input, $x$ , is unknown.
- $3x - 7 = 11$ .
Step 2: Isolate the variable
- Add $7$ to both sides: $3x = 11 + 7 = 18$ .
- Divide both sides by $3$ : $x = \dfrac{18}{3} = 6$ .
- Check: $g(6)$ $= 3 \cdot 6 - 7$ $= 18 - 7$ $= 11$ . It matches.

Traps

Losing the sign when squaring. $(-2)^2$ is $4$ , not $-4$ . Using $-4$ gives $2 \cdot (-4) + 6 + 1$ $= -8 + 6 + 1$ $= -1$ , a wrong answer.
Mixing up the input and the output. $g(x) = 11$ gives the output, so solve for $x$ . Working out $g(11)$ $= 3 \cdot 11 - 7$ $= 33 - 7$ $= 26$ answers a different question.

Practice this lesson32 questions

Unit 2: Nested functions

Key points

$g(f(x))$ means: work out $f$ first, then put its output into $g$ .
Work from the inside out.
The order matters. $g(f(x))$ and $f(g(x))$ are usually different.

See it solved

Example

The functions $f$ and $g$ are defined by $f(x) = x^2 + 1$ and $g(x) = 3x - 4$ . What is the value of $g(f(2))$ ?

Worked by hand

The plan

Work out the inner function
Plug the result into the outer function

Desmos: Desmos is quicker when the rules are given: define $f$ and $g$ , then type $g(f(2))$ .

Step 1: Work out the inner function
- The inner function is $f(2)$ .
- $f(2) = 2^2 + 1$ .
- $2^2 = 4$ , so $f(2) = 4 + 1 = 5$ .
Step 2: Plug the result into the outer function
- $g(f(2)) = g(5)$ .
- $g(5) = 3 \cdot 5 - 4$ .
- $3 \cdot 5 = 15$ , so $g(5)$ $= 15 - 4$ $= 11$ .

The correct answer is

11

The same example in Desmos

new graph
Keys: f(x)=x^2 → +1
Desmos shows: The first function.
Desmos shows: The second function.
Desmos shows: $11$ , at the right of the row.

Desmos expression list: f(x)=x^2+1; g(x)=3x-4; g(f(2)) — Type

Made with Desmos

Open Desmos

Your turn

For the functions $f$ and $g$ : $f(2) = 7$ , $f(5) = 3$ , $g(2) = 5$ and $g(7) = 1$ . What is the value of $g(f(2))$ ?

Show the answerHide the answer

Answer

The correct answer is $1$ .

Step 1: Work out the inner function
- The inner function is $f(2)$ . From the list, $f(2) = 7$ .
Step 2: Plug the result into the outer function
- $g(f(2)) = g(7)$ .
- From the list, $g(7) = 1$ .

Traps

Working from the outside in. $g(2)$ $= 3 \cdot 2 - 4$ $= 2$ , and then $f(2) = 5$ . That is $f(g(2)) = 5$ , the other order, a wrong answer.
Multiplying the two outputs. $g(f(2))$ is not $g(2) \cdot f(2)$ . $2 \cdot 5 = 10$ is a wrong answer.

Practice this lesson28 questions

Unit 3: Zeros and factors

Key points

A zero of a function is an input that gives an output of $0$ . On the graph it is an $x$ -intercept.
A zero at $x = n$ means $(x - n)$ is a factor. The sign flips: a zero at $-3$ gives the factor $(x + 3)$ .
If the graph crosses the $x$ -axis at a zero, the factor appears once. If the graph touches the axis and turns back, the factor is squared.
‘Distinct zeros’ counts each different zero once.

See it solved

Example

In the $xy$ -plane, the graph of the polynomial function $p$ crosses the $x$ -axis at $x = -2$ and at $x = 1$ . It touches the $x$ -axis at $x = 4$ without crossing it. Which equation could define $p$ ?

A $p(x)$ $= (x - 2)(x + 1)(x + 4)^2$
B $p(x)$ $= (x + 2)(x - 1)(x - 4)$
C $p(x)$ $= (x + 2)(x - 1)(x - 4)^2$
D $p(x)$ $= (x + 2)^2(x - 1)^2(x - 4)$

Worked by hand

The plan

Write a factor for each zero
Determine which factors are squared

Desmos: Reading the factors is quicker. In Desmos, type a choice and click its $x$ -intercepts to check.

Step 1: Write a factor for each zero
- A zero at $-2$ gives the factor $(x + 2)$ .
- A zero at $1$ gives the factor $(x - 1)$ .
- A zero at $4$ gives the factor $(x - 4)$ .
- That rules out choice A, which has every sign the wrong way round.
Step 2: Determine which factors are squared
- The graph crosses the axis at $-2$ and at $1$ . Those factors appear once.
- The graph touches the axis and turns back at $4$ . That factor is squared.
- $p(x)$ $= (x + 2)(x - 1)(x - 4)^2$ ⁠. That rules out choices B and D.

The correct answer choice is C:

p(x)

= (x + 2)(x - 1)(x - 4)^2

⁠.

The same example in Desmos

new graph
Desmos shows: The curve for choice C. Zoom out to see all of it.
click the points where the curve meets the x-axis
Desmos shows: $(-2, 0)$ , $(1, 0)$ and $(4, 0)$ . The curve crosses at the first two and only touches at $(4, 0)$ .

Desmos expression list: y=(x+2)(x-1)(x-4)^2 — Type

Desmos graph of y=(x+2)(x-1)(x-4)^2, with (−2, 0) and (1, 0) and (4, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $f$ is defined by $f(x)$ $= x(x - 3)(x + 5)^2$ . How many distinct zeros does $f$ have?

Show the answerHide the answer

Answer

The correct answer is $3$ .

Step 1: Set each factor equal to $0$
- $x = 0$ .
- $x - 3 = 0$ gives $x = 3$ .
- $(x + 5)^2 = 0$ gives $x = -5$ .
Step 2: Count the distinct zeros
- The zeros are $-5$ , $0$ and $3$ .
- The squared factor still gives only one zero, $-5$ . Count it once.
- There are $3$ distinct zeros.

Traps

Keeping the sign of the zero. A zero at $-2$ gives $(x + 2)$ , not $(x - 2)$ . Choice A has every sign the wrong way round.
Counting a squared factor twice. $(x + 5)^2$ gives the zero $-5$ once. Counting it twice gives $4$ distinct zeros, a wrong answer.

Practice this lesson24 questions

Unit 4: Moving and flipping graphs

Key points

$f(x) + c$ moves the graph up $c$ units. $f(x) - c$ moves it down $c$ units.
$f(x - c)$ moves the graph right $c$ units. $f(x + c)$ moves it left $c$ units. Inside the parentheses works backwards.
$-f(x)$ reflects the graph across the $x$ -axis. $f(-x)$ reflects it across the $y$ -axis.
A shift moves every point the same way, so you can just move the vertex.

See it solved

Example

The function $f$ is defined by $f(x)$ $= (x - 1)^2 + 2$ . The graph of $y = g(x)$ is the graph of $y = f(x)$ shifted $3$ units left and $5$ units down. Which equation defines $g$ ?

A $g(x)$ $= (x - 4)^2 - 3$
B $g(x)$ $= (x + 2)^2 - 3$
C $g(x)$ $= (x + 2)^2 + 7$
D $g(x)$ $= (x - 2)^2 - 3$

Worked by hand

The plan

Determine the translation
Plug the input into the function

Desmos: With answer choices, Desmos is quicker: define $f$ , type the shift as the question says it, then type each choice and keep the one that lies on top.

Step 1: Determine the translation
- $3$ units left: replace $x$ with $(x + 3)$ .
- $5$ units down: subtract $5$ at the end.
- So $g(x)$ $= f(x + 3) - 5$ .
Step 2: Plug the input into the function
- $f(x + 3)$ $= ((x + 3) - 1)^2 + 2$ .
- $3 - 1 = 2$ , so $f(x + 3)$ $= (x + 2)^2 + 2$ .
- Now subtract $5$ : $2 - 5 = -3$ .
- $g(x)$ $= (x + 2)^2 - 3$ .
- Check with the vertex. $(1, 2)$ moves left to $x = 1 - 3 = -2$ and down to $y = 2 - 5 = -3$ .
- The new vertex is $(-2, -3)$ , which matches $(x + 2)^2 - 3$ .

The correct answer choice is B:

g(x)

= (x + 2)^2 - 3

The same example in Desmos

new graph
Keys: f(x)=(x-1)^2 → +2
Desmos shows: The original parabola.
Desmos shows: A second parabola, to the left of the first and lower.
Keys: y=(x+2)^2 → -3
Desmos shows: Choice B lies exactly on top of $g$ (dashed in the picture).
click the lowest point of the new parabola
Desmos shows: $(-2, -3)$ : the new vertex.

Desmos expression list: f(x)=(x-1)^2+2; g(x)=f(x+3)-5; y=(x+2)^2-3 — Type

Desmos graph of f(x)=(x-1)^2+2; g(x)=f(x+3)-5; y=(x+2)^2-3, with (−2, −3) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $f$ is defined by $f(x)$ $= x^2 - 4x + 1$ . The graph of $y = g(x)$ is the graph of $y = f(x)$ reflected across the $x$ -axis. What is the value of $g(3)$ ?

Show the answerHide the answer

Answer

The correct answer is $2$ .

Step 1: Determine the translation
- A reflection across the $x$ -axis changes the sign of every output: $g(x) = -f(x)$ .
Step 2: Plug the input into the function
- $f(3)$ $= 3^2 - 4 \cdot 3 + 1$ .
- $3^2 = 9$ and $4 \cdot 3 = 12$ .
- $f(3)$ $= 9 - 12 + 1$ $= -2$ .
- $g(3)$ $= -f(3)$ $= -(-2)$ $= 2$ .

Traps

Moving the wrong way inside the parentheses. ‘ $3$ left’ is $(x + 3)$ , not $(x - 3)$ . Using $x - 3$ gives $(x - 4)^2 - 3$ , choice A, a wrong answer.
Adding for ‘down’. Down means subtract. $2 + 5 = 7$ gives $(x + 2)^2 + 7$ , choice C, a wrong answer.

Practice this lesson68 questions

Unit 5: Remainders and the factor theorem

Key points

When a polynomial $p(x)$ is divided by $x - n$ , the remainder is $p(n)$ .
For a divisor $x + n$ , plug in $x = -n$ .
If $p(n) = 0$ , the remainder is $0$ . Then $x - n$ is a factor of $p(x)$ .
No long division is needed. Just plug in.

See it solved

Example

The polynomial $p$ is defined by $p(x)$ $= x^3 - 4x^2 + 11$ . What is the remainder when $p(x)$ is divided by $x - 3$ ?

Worked by hand

The plan

Determine the input
Plug the input into the function

Desmos: Desmos is quicker with a long polynomial: define $p$ once, then type $p(3)$ .

Step 1: Determine the input
- The divisor is $x - 3$ . It equals $0$ when $x = 3$ .
- So the remainder is $p(3)$ .
Step 2: Plug the input into the function
- $p(3)$ $= 3^3 - 4 \cdot 3^2 + 11$ .
- $3^3 = 27$ .
- $3^2 = 9$ , so $4 \cdot 9 = 36$ .
- $p(3)$ $= 27 - 36 + 11$ $= 2$ . The remainder is $2$ .

The correct answer is

2

The same example in Desmos

new graph
Keys: p(x)=x^3 → -4x^2 → +11
Desmos shows: The polynomial.
Desmos shows: $2$ , at the right of the row. The remainder is $2$ .

Desmos expression list: p(x)=x^3-4x^2+11; p(3) — Type

Desmos graph of p(x)=x^3-4x^2+11; p(3), with (3, 2) labelled — Look

Made with Desmos

Open Desmos

Your turn

The polynomial $p$ is defined by $p(x)$ $= x^3 + kx - 6$ , where $k$ is a constant. If $x - 2$ is a factor of $p(x)$ , what is the value of $k$ ?

Show the answerHide the answer

Answer

The correct answer is $-1$ .

Step 1: Set $p(2)$ equal to $0$
- $x - 2$ is a factor, so the remainder $p(2)$ is $0$ .
- $p(2)$ $= 2^3 + k \cdot 2 - 6$ .
- $2^3 = 8$ , so $p(2)$ $= 8 + 2k - 6$ .
- $8 - 6 = 2$ , so $p(2) = 2 + 2k$ .
Step 2: Solve for $k$
- $2 + 2k = 0$ .
- Subtract $2$ from both sides: $2k = -2$ .
- Divide both sides by $2$ : $k = -1$ .

Traps

Plugging in the wrong sign. For the divisor $x - 3$ , plug in $3$ , not $-3$ . $p(-3)$ $= -27 - 36 + 11$ $= -52$ is a wrong answer.
Answering with the constant term. $11$ is $p(0)$ , the remainder when dividing by $x$ . Here the divisor is $x - 3$ , so the remainder is $p(3)$ .

Practice this lesson28 questions

Unit 6: Where a function is undefined

Key points

A fraction is undefined when its denominator is $0$ .
To find where, set the denominator equal to $0$ and solve.
The numerator does not matter for this.
On the graph, the curve breaks there: it has a vertical asymptote.

See it solved

Example

The function $f$ is defined by $f(x)$ $= \dfrac{x + 2}{x^2 - 5x + 6}$ . What is the greatest value of $x$ for which $f(x)$ is undefined?

Worked by hand

The plan

Set the denominator to $0$
Solve the equation

Desmos: Desmos is quicker here: type the denominator set equal to $0$ and read the vertical lines.

Step 1: Set the denominator to $0$
- $x^2 - 5x + 6 = 0$ .
Step 2: Solve the equation
- Find two numbers that multiply to $6$ and add to $-5$ : they are $-2$ and $-3$ .
- $(x - 2)(x - 3)$ $= 0$ .
- $x = 2$ or $x = 3$ .
- $f$ is undefined at $2$ and at $3$ . The greatest is $3$ .
- Check: $3^2 - 5 \cdot 3 + 6$ $= 9 - 15 + 6$ $= 0$ .

The correct answer is

3

The same example in Desmos

new graph
Keys: x^2 → -5x+6=0
Desmos shows: Two vertical lines.
click where each line meets the x-axis
Desmos shows: $(2, 0)$ and $(3, 0)$ . $f$ is undefined at $x = 2$ and at $x = 3$ .

Desmos expression list: x^2-5x+6=0 — Type

Desmos graph of x^2-5x+6=0, with (2, 0) and (3, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $h$ is defined by $h(x) = \dfrac{3x}{2x - 10}$ . For what value of $x$ is $h(x)$ undefined?

Show the answerHide the answer

Answer

The correct answer is $5$ .

Step 1: Set the denominator to $0$
- $2x - 10 = 0$ .
Step 2: Solve the equation
- Add $10$ to both sides: $2x = 10$ .
- Divide both sides by $2$ : $x = \dfrac{10}{2} = 5$ .

Traps

Using the numerator. $x + 2 = 0$ gives $x = -2$ , but that is where $f(x) = 0$ , not where it is undefined. $-2$ is a wrong answer.
Stopping at one factor. $(x - 2)(x - 3)$ $= 0$ has two solutions. Giving only $2$ misses the greatest value, $3$ .

Practice this lesson24 questions

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
Type y1 and Desmos shows y₁: a digit typed straight after a letter becomes a subscript. If you type the underscore yourself (y_1), press → after the digit: a ~ typed next lands inside the subscript (+, − and = leave it by themselves).
The ~ key is Shift + ` on most keyboards (the key left of 1). Use ~, not =: ~ asks Desmos to find the unknown letters; = only draws or defines.
Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
Start a new graph for each example: a letter or function defined twice gives the error 'You've defined … in more than one place'.

Start practicing

Practice by lesson

Lesson 1, Function notation, tables and graphs, 32 questions
Lesson 2, Nested functions, 28 questions
Lesson 3, Zeros and factors, 24 questions
Lesson 4, Moving and flipping graphs, 68 questions
Lesson 5, Remainders and the factor theorem, 28 questions
Lesson 6, Where a function is undefined, 24 questions

Mixed practice: every lesson

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11 questions

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Next chapter · Chapter 6Quadratics and parabolas

Desmos

Next set

1Unit 1: Function notation, tables and graphs

Key points

See it solved

Worked by hand

Step 1: Plug the input into the function

The same example in Desmos

Your turn

Step 1: Set the function equal to the output

Step 2: Isolate the variable

Traps

2Unit 2: Nested functions

Key points

See it solved

Worked by hand

Step 1: Work out the inner function

Step 2: Plug the result into the outer function

The same example in Desmos

Your turn

Step 1: Work out the inner function

Step 2: Plug the result into the outer function

Traps

3Unit 3: Zeros and factors

Key points

See it solved

Worked by hand

Step 1: Write a factor for each zero

Step 2: Determine which factors are squared

The same example in Desmos

Your turn

Step 1: Set each factor equal to 000

Step 2: Count the distinct zeros

Traps

4Unit 4: Moving and flipping graphs

Key points

See it solved

Worked by hand

Step 1: Determine the translation

Step 2: Plug the input into the function

The same example in Desmos

Your turn

Step 1: Determine the translation

Step 2: Plug the input into the function

Traps

5Unit 5: Remainders and the factor theorem

Key points

See it solved

Worked by hand

Step 1: Determine the input

Step 2: Plug the input into the function

The same example in Desmos

Your turn

Step 1: Set p(2)p(2)p(2) equal to 000

Step 2: Solve for kkk

Traps

6Unit 6: Where a function is undefined

Key points

See it solved

Worked by hand

Step 1: Set the denominator to 000

Step 2: Solve the equation

The same example in Desmos

Your turn

Step 1: Set the denominator to 000

Step 2: Solve the equation

Traps

Practice by lesson

Mixed practice: every lesson

Easy set 1

800-level question

What tripped you up

Unit 1: Function notation, tables and graphs

Unit 2: Nested functions

Unit 3: Zeros and factors

Step 1: Set each factor equal to $0$

Unit 4: Moving and flipping graphs

Unit 5: Remainders and the factor theorem

Step 1: Set $p(2)$ equal to $0$

Step 2: Solve for $k$

Unit 6: Where a function is undefined

Step 1: Set the denominator to $0$

Step 1: Set the denominator to $0$