MPM2D · Unit 4 test review

Quadratic Relations: Test Review

Everything for the Unit 4 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Quadratic Relations and Their Properties

A quadratic relation can be written as $y = a x^{2} + b x + c$ with $a \neq = 0$ . Its graph is a parabola.
With equal steps in $x$ : constant first differences mean linear; constant, non-zero second differences mean quadratic.
When $x$ goes up by 1 each step, the second difference is $2 a$ . Positive opens up (minimum); negative opens down (maximum).
The axis of symmetry $x = h$ passes through the vertex and lies halfway between any two points with the same $y$ -value, such as the two zeros.
The maximum or minimum value is the $y$ -coordinate of the vertex. A parabola has 0, 1 or 2 zeros; $y = a x^{2} + b x + c$ has $y$ -intercept $c$ .

Transformations and Vertex Form

$y = a (x - h)^{2} + k$ has vertex $(h, k)$ and axis of symmetry $x = h$ .
$k$ moves the graph up ( $k > 0$ ) or down ( $k < 0$ ). $h$ moves it right ( $h > 0$ ) or left ( $h < 0$ ): $y = (x + 3)^{2}$ has $h = - 3$ .
$a < 0$ reflects in the x-axis (opens down). $a$ beyond $1$ or $- 1$ stretches vertically; $a$ between $- 1$ and $1$ (not 0) compresses vertically.
From the vertex, the step pattern 1, 3, 5 of $y = x^{2}$ becomes $a$ , $3 a$ , $5 a$ .
Equation from a graph: read $(h, k)$ , substitute one other point into $y = a (x - h)^{2} + k$ , and solve for $a$ .

Factored Form and Zeros

Zero product property: if $A \times B = 0$ , then $A = 0$ or $B = 0$ .
$y = a (x - r) (x - s)$ has zeros $x = r$ and $x = s$ ; the factor $(x + 2)$ gives the zero $x = - 2$ .
The axis of symmetry is halfway between the zeros: $x = \frac{r + s}{2}$ . Substitute it to get the vertex.
The $y$ -intercept of $y = a (x - r) (x - s)$ is $a \cdot r \cdot s$ .
To find $a$ from the zeros, substitute one more point into $y = a (x - r) (x - s)$ and solve.

Completing the Square

$(x + d)^{2} = x^{2} + 2 d x + d^{2}$ : the constant is the square of half the coefficient of $x$ .
$x^{2} + b x = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}$ . Add and subtract the same number, so the relation is unchanged.
If $a \neq = 1$ , factor $a$ from the $x^{2}$ and $x$ terms only. When the subtracted square leaves the brackets, multiply it by $a$ .
Partial factoring: $y = x (a x + b) + c$ equals $c$ at $x = 0$ and $x = - \frac{b}{a}$ , so the axis is halfway between them.
Check every completed square by expanding it back to standard form.

Comparing y = x squared and y = 2 to the x

$y = x^{2}$ has constant second differences; $y = 2^{x}$ has a constant ratio: each $y$ -value is double the one before.
$y = 2^{x}$ has $y$ -intercept 1, no zeros, is always increasing, and has the x-axis as a horizontal asymptote.
$y = x^{2}$ and $y = 2^{x}$ meet at $(2, 4)$ and $(4, 16)$ and once between $x = - 1$ and $x = 0$ ; for $x > 4$ , $2^{x} > x^{2}$ .
$a^{0} = 1$ for any $a \neq = 0$ .
$a^{- n} = \frac{1}{a ^{n}}$ for any $a \neq = 0$ : a negative exponent gives a reciprocal, not a negative number.

Where marks get lost

Using finite differences when the $x$ -values do not go up by equal steps, for example $x = 0, 1, 3, 4$ . The test only works with equal steps. (Quadratic Relations and Their Properties)
Mixing up the order of subtraction. Use later minus earlier every time: if you take the first differences earlier minus later but the second differences later minus earlier, every second difference changes sign and an upward parabola looks like it opens down. (Quadratic Relations and Their Properties)
Giving the vertex when asked for the minimum value. For vertex $(3, - 4)$ the minimum value is $- 4$ , a single number. (Quadratic Relations and Their Properties)
Writing the axis of symmetry as " $3$ " or " $y = 3$ ". It is a vertical line, so its equation is $x = 3$ . (Quadratic Relations and Their Properties)
Assuming every parabola has two zeros. $y = x^{2} + 1$ has none (its lowest point is $(0, 1)$ ), and $y = x^{2}$ has exactly one. (Quadratic Relations and Their Properties)
Reading $h$ with the wrong sign. In $y = (x + 4)^{2}$ the graph moves left 4 and the vertex is $(- 4, 0)$ , because $x + 4 = x - (- 4)$ . (Transformations and Vertex Form)
Thinking $y = \frac{1}{2} x^{2}$ is narrower than $y = x^{2}$ . Multiplying the $y$ -values by $\frac{1}{2}$ is a vertical compression, so the graph is wider. (Transformations and Vertex Form)
Stepping 1, 3, 5 from the vertex without multiplying by $a$ . For $y = 3 (x - 1)^{2}$ the steps are 3, 9, 15. (Transformations and Vertex Form)
Treating $k$ as the $y$ -intercept. For $y = 2 (x - 3)^{2} + 1$ the $y$ -intercept is $2 (9) + 1 = 19$ ; $k = 1$ is the minimum value. (Transformations and Vertex Form)
When finding $a$ , forgetting to square the bracket: $5 = a (4 - 2)^{2} - 3$ gives $5 = 4 a - 3$ , not $5 = 2 a - 3$ . (Transformations and Vertex Form)
Reading zeros with the wrong sign. $y = (x + 3) (x - 1)$ has zeros $- 3$ and $1$ , not $3$ and $- 1$ . (Factored Form and Zeros)
Reading the zero of $(2 x - 5)$ as $5$ . Solve $2 x - 5 = 0$ : $x = \frac{5}{2} = 2.5$ . (Factored Form and Zeros)
Losing the zero at $x = 0$ . $y = 2 x (x + 3)$ has zeros $0$ and $- 3$ ; the factor $x$ counts. (Factored Form and Zeros)
Leaving out $a$ when building an equation from zeros. $y = (x + 1) (x - 5)$ has the right zeros but a $y$ -intercept of $- 5$ , not $10$ . (Factored Form and Zeros)
Dropping the leading factor when finding the vertex. For $y = 2 (x - 5) (x + 3)$ the vertex value is $2 (- 4) (4) = - 32$ , not $(- 4) (4) = - 16$ . (Factored Form and Zeros)
Forgetting to multiply the subtracted square by $a$ . In $2 (x^{2} + 4 x + 4 - 4) + 1$ , the $- 4$ leaves as $- 8$ , giving $2 (x + 2)^{2} - 7$ , not $2 (x + 2)^{2} - 3$ . (Completing the Square)
Halving $b$ instead of $\frac{b}{a}$ . For $3 x^{2} - 12 x$ , the bracket is $x^{2} - 4 x$ , so half of $- 4$ (which is $- 2$ ) is used, not half of $- 12$ . (Completing the Square)
Factoring $a$ out of the constant too, which creates fractions: $3 (x^{2} - 4 x + \frac{1}{3})$ . Factor $a$ from the first two terms only. (Completing the Square)
Adding the square without subtracting it. Writing $x^{2} + 6 x + 4 = (x + 3)^{2} + 4$ adds 9 to the relation and moves the parabola. (Completing the Square)
Misreading the final vertex. $(x + 4)^{2} - 6$ has vertex $(- 4, - 6)$ , not $(4, - 6)$ . (Completing the Square)
Making a negative exponent negative: $2^{- 3} = - 8$ or $- 6$ . It is $\frac{1}{2 ^{3}} = \frac{1}{8}$ . (Comparing y = x squared and y = 2 to the x)
Writing $5^{0} = 0$ or $5^{0} = 5$ . Any non-zero base to the exponent 0 is $1$ . (Comparing y = x squared and y = 2 to the x)
Ignoring brackets: $- 3^{0} = - 1$ but $(- 3)^{0} = 1$ , because without brackets the exponent applies only to the 3. (Comparing y = x squared and y = 2 to the x)
Mixing up $2^{x}$ , $2 x$ and $x^{2}$ . At $x = 5$ : $2^{5} = 32$ , $2 (5) = 10$ and $5^{2} = 25$ . (Comparing y = x squared and y = 2 to the x)
Concluding from a few small values that $x^{2}$ grows faster. It is ahead at $x = 3$ ( $9$ versus $8$ ), but from $x = 5$ on, $2^{x}$ is always bigger. (Comparing y = x squared and y = 2 to the x)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Quadratic Relations and Their Properties
1
Each table uses $x = 0, 1, 2, 3$ . Which list of $y$ -values comes from a quadratic relation?
Quadratic Relations and Their Properties
2
A parabola opens down and its vertex is $(4, - 2)$ . Which statement is true?
Quadratic Relations and Their Properties
3
A parabola has vertex $(3, - 4)$ and one zero at $x = 1$ . What is the other zero?
Quadratic Relations and Their Properties
4
How many zeros can a parabola have?
Quadratic Relations and Their Properties
5
Match each feature of a parabola with its description.
Vertex
Axis of symmetry
Zero
y-intercept
Transformations and Vertex Form
6
What is the vertex of $y = 3 (x + 4)^{2} - 1$ ?
Transformations and Vertex Form
7
What is the $y$ -intercept of $y = - 2 (x - 1)^{2} + 7$ ?
Transformations and Vertex Form
8
Which statement describes $y = - 3 (x - 2)^{2} + 5$ ?
Transformations and Vertex Form
9
How many zeros does $y = 2 (x - 1)^{2} + 3$ have?
Transformations and Vertex Form
10
Put the steps for sketching $y = a (x - h)^{2} + k$ by hand in order.
1. 1Plot the vertex (h, k) and draw the axis of symmetry x = h.
2. 2Join the plotted points with a smooth curve.
3. 3From the vertex, step 1 unit left and right at a time, moving vertically a, then 3a, then 5a.
4. 4Read the values of a, h and k from the equation.
Factored Form and Zeros
11
What are the zeros of $y = 3 (x - 2) (x + 7)$ ?
Factored Form and Zeros
12
What is the $y$ -intercept of $y = - 2 (x - 1) (x + 4)$ ?
Factored Form and Zeros
13
Match each relation with its zeros.
1. A. $- 6$ and $1$
2. B. $0$ and $6$
3. C. $6$ and $- 1$
4. D. $6$ only
$y = (x - 6) (x + 1)$
$y = (x + 6) (x - 1)$
$y = x (x - 6)$
$y = (x - 6)^{2}$
Factored Form and Zeros
14
How many zeros does $y = (x - 4)^{2}$ have?
Factored Form and Zeros
15
Put the steps for sketching $y = x^{2} - 6 x + 8$ from its factors in order.
1. 1Plot the zeros, vertex and y-intercept, then draw a smooth curve.
2. 2Substitute x = 3 to get the vertex (3, −1).
3. 3Average the zeros to get the axis of symmetry x = 3.
4. 4Factor the expression as (x − 2)(x − 4).
5. 5Set each factor equal to zero to get the zeros 2 and 4.
Completing the Square
16
What number must be added to $x^{2} + 14 x$ to make a perfect square trinomial?
Completing the Square
17
Find $k$ if $2 x^{2} + 12 x + 5 = 2 (x + 3)^{2} + k$ .
Completing the Square
18
Put the steps for writing $y = 2 x^{2} + 8 x - 1$ in vertex form in order.
1. 1Add and subtract 4 inside the brackets: 2(x² + 4x + 4 − 4) − 1
2. 2Write the perfect square and combine constants: 2(x + 2)² − 9
3. 3Move the −4 out by multiplying it by 2: 2(x² + 4x + 4) − 8 − 1
4. 4Factor 2 from the first two terms: 2(x² + 4x) − 1
Completing the Square
19
A classmate wrote: $2 x^{2} + 8 x + 1 = 2 (x^{2} + 4 x + 4 - 4) + 1 = 2 (x + 2)^{2} - 3 .$ What went wrong?
Completing the Square
20
A ball's height in metres after $t$ seconds is $h = - 5 t^{2} + 30 t + 1$ . Complete the square to find its maximum height.
m
Comparing y = x squared and y = 2 to the x
21
Evaluate $2^{- 4}$ . Give a fraction or an exact decimal.
Comparing y = x squared and y = 2 to the x
22
What is the value of $- 5^{0}$ ?
Comparing y = x squared and y = 2 to the x
23
Write $1 0^{- 3}$ as a decimal.
Comparing y = x squared and y = 2 to the x
24
Match each description with its value.
$y$ -intercept of $y = x^{2}$
$y$ -intercept of $y = 2^{x}$
value of $x^{2}$ at $x = - 2$
value of $2^{x}$ at $x = - 2$
Comparing y = x squared and y = 2 to the x
25
For $y = 2^{x}$ with $x = 0, 1, 2, 3, 4$ , what happens to the second differences?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…