VertexMath · MPM2D · Free worksheet

Introduction to Quadratic Relations Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Second differences: when $x$ goes up by $1$ each time, constant second differences (not $0$ ) mean the relation is quadratic, and the second difference equals $2a$ . Positive: opens up (minimum). Negative: opens down (maximum).
Find differences as next minus previous and keep the signs. Turning the differences positive hides a quadratic pattern.
Factored form $y=a(x-r)(x-s)$ : the zeros are $r$ and $s$ , so $(x+5)$ gives the zero $x=-5$ . If $a>0$ the parabola opens up; if $a<0$ it opens down.
Vertex: the axis of symmetry is halfway between the zeros, $x=\dfrac{r+s}{2}$ . Substitute that $x$ into the equation to get the maximum or minimum value.
Expanding: $(x+b)^2=x^2+2bx+b^2$ , never $x^2+b^2$ . Multiply the two brackets first, then multiply by $a$ .

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Second Differences · Properties · Factored Form · Expanding

K1 Linear, Quadratic or Neither?

For each table, determine the first and second differences. Then state whether the relation is linear, quadratic or neither. If it is quadratic, state the direction of opening.

a)
Table A

$x$ $0$ $1$ $2$ $3$ $4$
$y$ $6$ $9$ $10$ $9$ $6$
b)
Table B

$x$ $-2$ $-1$ $0$ $1$ $2$
$y$ $-5$ $-2$ $1$ $4$ $7$
c)
Table C

$x$ $0$ $1$ $2$ $3$ $4$
$y$ $1$ $3$ $9$ $27$ $81$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-5$	$-2$	$1$	$4$	$7$

Stuck? Peek at a hint

Subtract each $y$ -value from the next one and keep the sign. Then do the same with the first differences.

K2 Read the Parabola

Use the graph below. Each grid square is 1 unit.

a)
State the coordinates of the vertex and the equation of the axis of symmetry.
b)
State the zeros and the $y$ -intercept.
c)
State whether the relation has a maximum or a minimum value, and state that value.

Stuck? Peek at a hint

The vertex is the turning point. The axis of symmetry is a vertical line, so its equation starts with $x=$ .

K3 Symmetry in a Table

The table shows a quadratic relation.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-4$	$2$	$6$	$8$	$8$	$6$

a)
Determine the equation of the axis of symmetry.
b)
Use symmetry to determine $y$ when $x=3$ and when $x=4$ .
c)
The relation is $y=-x^2+x+8$ . Determine the coordinates of the vertex and the maximum value.

Stuck? Peek at a hint

Find two $x$ -values with the same $y$ -value. The axis of symmetry is halfway between them.

K4 Factored Form

a)
For $y=2(x-3)(x+5)$ , state the zeros, the direction of opening and the $y$ -intercept.
b)
Determine the equation of the axis of symmetry and the vertex of $y=2(x-3)(x+5)$ .
c)
For $y=-x(x-6)$ , state the zeros and determine the vertex.

Stuck? Peek at a hint

Set each factor equal to $0$ and solve. The vertex lies on the axis of symmetry, halfway between the zeros.

K5 Expand to Standard Form

a)
Complete the area model, then expand and simplify $(x+6)(x-2)$ .

$\times$ $x$ $-2$
$x$
$6$
b)
Expand and simplify $(2x-3)^2$ .
c)
Write $y=-3(x-4)(x+1)$ in standard form.

$\times$	$x$	$-2$
$x$
$6$

Stuck? Peek at a hint

Multiply every term in the first bracket by every term in the second. In (c), expand the two brackets before multiplying by $-3$ .

Real World (A)

Optimization · Revenue · Modelling Data

A1 Garden by the River

A community garden along the Grand River is fenced into two equal rectangular plots using $60$ m of fencing, as shown. The side along the river needs no fence. Let $w$ be the width and $L$ the total length along the river, both in metres.

a)
Write $L$ in terms of $w$ . Then write the total area $A(w)$ in factored form.
b)
Determine the dimensions that give the maximum total area, and state that area.
c)
The flat stretch of riverbank is only $24$ m long, so $L\le 24$ . Determine the largest possible total area now.

Stuck? Peek at a hint

Count the fence pieces: three widths and one length use all $60$ m.

A2 Climbing Gym Passes

A climbing gym in Ottawa sells $100$ day passes a week at $40 each. For every $1 increase in price, it sells $5$ fewer passes; for every $1 decrease, $5$ more. Let $x$ be the number of $1 increases, so a negative $x$ is a decrease.

a)
Write the weekly revenue $R(x)$ in factored form and state its zeros.
b)
What does each zero mean for the gym?
c)
Determine the vertex. What price maximizes weekly revenue, and what is the maximum revenue?

Stuck? Peek at a hint

Revenue is price times passes sold: write each one as an expression in $x$ .

A3 Tracking a Kick

A drone films a soccer ball kicked from the ground. The table and scatter plot show its height $h$ , in metres, $t$ seconds after the kick. The ball lands at about $t=3$ s.

$t\ \text{(s)}$	$0$	$0.5$	$1.0$	$1.5$	$2.0$	$2.5$	$3.0$
$h\ \text{(m)}$	$0$	$6.2$	$9.9$	$11.2$	$9.8$	$6.3$	$0.1$

a)
Use the zeros and the point $(1.5,\,11.2)$ to determine a model $h=at(t-3)$ . Round $a$ to the nearest whole number.
b)
Use your model to determine the height at $t=0.8$ s. At what other time is the ball at that height?
c)
Use your model to predict the height at $t=3.4$ s. Is the prediction realistic?

Stuck? Peek at a hint

Substitute $t=1.5$ and $h=11.2$ into $h=at(t-3)$ , then solve for $a$ .

Think It Through (T)

Error Analysis · Building Equations

T1 Spot the Flaw

A student analyzes the table below.
"The second differences are all $-4$ , so $a=-4$ and the parabola opens down. The $y$ -intercept is $3$ , so the vertex is $(0,3)$ ."

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$3$	$5$	$3$	$-3$	$-13$

a)
Determine the second differences of $y=-4x^2$ for $x=0,1,2,3$ . What value of $a$ gives a second difference of $-4$ ?
b)
Use the table to determine the vertex. What is the point $(0,3)$ ?
c)
Use only the values at $x=0$ , $2$ and $4$ (steps of $2$ ). Determine the second difference. Does it still equal $2a$ ?

Stuck? Peek at a hint

For (a), make a table for $y=-4x^2$ and take its differences. For (b), look for two $x$ -values with the same $y$ -value.

T2 Build It Backwards

Write an equation in factored form for each parabola.

a)
Zeros $-2$ and $6$ , and $y$ -intercept $6$ .
b)
Vertex $(3,27)$ , and one zero at $x=0$ .
c)
The parabola graphed below. Each grid square is 1 unit.

Stuck? Peek at a hint

Write $y=a(x-r)(x-s)$ with the zeros in place, then substitute one more known point to find $a$ .

Boss Level

Optional challenge

Chords of a Parabola

Consider the parabola $y=(x-2)(x-8)$ and the horizontal line $y=k$ .

a)
The line $y=k$ meets the parabola at two points that are $10$ units apart. Find $k$ .
b)
For which value of $k$ does the line meet the parabola at exactly one point?
c)
For which value of $a$ does the line $y=16$ meet $y=a(x-2)(x-8)$ at two points that are $4$ units apart?

Stuck? Peek at a hint

The two points are mirror images across the axis of symmetry. Find that axis first.

Answer key

Final answers only. Try first, then check.

Core Skills

a) First: $3,1,-1,-3$ ; second: $-2,-2,-2$ ; quadratic, opens down
b) First: $3,3,3,3$ ; second: $0,0,0$ ; linear
c) First: $2,6,18,54$ ; second: $4,12,36$ ; neither

a) Vertex $(3,-4)$ ; axis $x=3$
b) Zeros $1$ and $5$ ; $y$ -intercept $5$
c) Minimum value $-4$

a) $x=\frac{1}{2}$
b) $x=3$ : $y=2$ ; $x=4$ : $y=-4$
c) Vertex $\left(\frac{1}{2},\frac{33}{4}\right)=(0.5,\,8.25)$ ; maximum value $8.25$

a) Zeros $3$ and $-5$ ; opens up; $y$ -intercept $-30$
b) $x=-1$ ; vertex $(-1,-32)$
c) Zeros $0$ and $6$ ; vertex $(3,9)$

a) $x^2+4x-12$
b) $4x^2-12x+9$
c) $y=-3x^2+9x+12$

Real World

a) $L=60-3w$ ; $A(w)=w(60-3w)=-3w(w-20)$
b) $w=10$ m, $L=30$ m; $300$ m²
c) $w=12$ m, $L=24$ m; $288$ m² (the vertex $w=10$ would need $L=30$ m)

a) $R(x)=(40+x)(100-5x)=-5(x+40)(x-20)$ ; zeros $x=-40$ and $x=20$
b) $x=20$ : price $60, no passes sold. $x=-40$ : price $0, passes given away. Both give $0 revenue.
c) Vertex $(-10,4500)$ : lower the price by $10 to $30; $150$ passes; $4500 per week

a) $a\approx-4.98$ , so $h=-5t(t-3)$
b) $8.8$ m; $t=2.2$ s
c) $-6.8$ m. Not realistic: the ball landed at $t\approx3$ s, so the model only applies for $0\le t\le 3$ .

Think It Through

a) $-8$ and $-8$ ; $a=-2$
b) Vertex $(1,5)$ , since $y=3$ at both $x=0$ and $x=2$ ; $(0,3)$ is the $y$ -intercept
c) $-16$ . No: with steps of $2$ it is $4\times 2a=8a$

a) $y=-\frac{1}{2}(x+2)(x-6)$
b) $y=-3x(x-6)$
c) $y=2(x+3)(x-1)$

Boss Level

Boss

a) $k=16$ (the points are $(0,16)$ and $(10,16)$ )
b) $k=-9$
c) $a=-\frac{16}{5}$

Free cheat sheet

Quadratic Relations Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

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Phone card

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↓ Phone card (image)

Most testedFor $y=2(x-1)(x+5)$ , state the zeros, the axis of symmetry and the vertex. (card 3)

1Finite differences

UseDifference $=$ next $-$ previous; keep the signs. Second difference $=2a$ : positive opens up, negative opens down.

With $x$ going up by $1$ :

first differences constant	linear
second differences constant, not $0$	quadratic
neither constant	neither

Trap $y$ : $5,8,9,8,5$ has first differences $3,1,-1,-3$ , not $3,1,1,3$ .

TrapSecond difference $-4$ means $2a=-4$ , so $a=-2$ , not $-4$ .

2Parts of a parabola

UseVertex $(h,k)$ ; axis of symmetry $x=h$ ; the maximum or minimum value is $k$ . Zeros: where $y=0$ .

TrapVertex $(1,-4)$ : the minimum value is $-4$ and the axis is $x=1$ , not $y=1$ or the point.

Trap $y$ is equal at $x=0$ and $x=1$ , so the axis is $x=0.5$ : the vertex need not be in the table.

3Factored form

$y=a(x-r)(x-s)$

axis $x=\dfrac{r+s}{2}$

UseZeros $r$ and $s$ . $a>0$ opens up; $a<0$ opens down. Vertex: substitute the axis value.

Example $y=2(x-1)(x+5)$ : zeros $1$ , $-5$ ; axis $x=-2$ ; $y=2(-3)(3)=-18$ , so vertex $(-2,-18)$ .

Trap $(x+5)$ gives the zero $-5$ , not $5$ : solve $x+5=0$ .

TrapThe lone factor $x$ in $y=3x(x-4)$ is a zero too: $x=0$ .

4Finding $a$ ; best fit

UseFrom the zeros, write $y=a(x-r)(x-s)$ ; substitute one more point to solve for $a$ . A curve of best fit follows a scatter plot's trend.

ExampleZeros $-1$ and $4$ , through $(0,8)$ : $8=a(1)(-4)$ , so $a=-2$ .

TrapSolve for $a$ : it is not $1$ , the $y$ -intercept or the maximum height.

TrapInterpolate inside the data; extrapolating past it (after the ball lands) can give nonsense.

5Expanding

$(a+b)^2=a^2+2ab+b^2$

$(a+b)(a-b)=a^2-b^2$

UseMultiply the two brackets first (every term by every term, or an area model), then multiply by the constant in front.

Example $-2(x-3)(x+1)=-2(x^2-2x-3)=-2x^2+4x+6$ .

Trap $(3x-2)^2=9x^2-12x+4$ , not $9x^2+4$ : write it as $(3x-2)(3x-2)$ .

6Maximum and minimum

UseWrite the quantity in factored form. Zeros, then the axis, then substitute: the maximum is the $y$ -value. Answer the question asked.

Example $40$ m of fence, three sides, a wall on the fourth: $A=w(40-2w)$ , zeros $0$ and $20$ , so $w=10$ m and $A=200$ m $^2$ .

TrapA vertex at $x=-10$ price increases means lowering the price by $10; don't flip the sign.

7Zeros in context

UseA zero is where the quantity is $0$ : the ball is on the ground, or no revenue. Keep only values that make sense, like $t\ge0$ .

Example $h=-5t(t-4)$ : zeros $t=0$ (kick) and $t=4$ s (landing); maximum $h(2)=20$ m.

Where to start?

A table of values Differences

An equation in factored form Zeros, then axis

A greatest area or revenue Vertex

y

-value

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Core Skills (K)

K1 Linear, Quadratic or Neither?

K2 Read the Parabola

K3 Symmetry in a Table

K4 Factored Form

K5 Expand to Standard Form

Real World (A)

A1 Garden by the River

A2 Climbing Gym Passes

A3 Tracking a Kick

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

Chords of a Parabola

Answer key

Core Skills

Real World

Think It Through

Boss Level

Quadratic Relations Cheat Sheet

Unit Summary

Build your own

Phone card

1Finite differences

2Parts of a parabola

3Factored form

4Finding aaa; best fit

5Expanding

6Maximum and minimum

7Zeros in context

Where to start?

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What is VertexMath?

4Finding $a$ ; best fit