A. Quadratic Functions · Quadratic Functions

Sketching from Standard Form: Choosing a Strategy

A2.11 A2.10About 70 minutes

Learning goals

I can choose a strategy to find the vertex of f(x) = ax² + bx + c.
I can use partial factoring and h = −b ÷ (2a).
I can sketch a quadratic function from standard form.
I can list all the key features of the graph.

Words to know

partial factoring
Factoring x out of the first two terms, f(x) = x(ax + b) + c, to find two points with the same y-value c.

h = −b ÷ (2a)
The x-coordinate of the vertex (the axis of symmetry) of f(x) = ax² + bx + c.

key features
Direction, vertex, axis, max or min value, intercepts, domain and range.

mirror point
The point on the other side of the axis of symmetry with the same y-value.

The form you are usually given

In the real world, quadratic models often arrive in standard form, because that is what comes out of physics formulas and spreadsheets: h(t) = −4.9t² + 15t + 2, or P(x) = −2x² + 80x − 300. To sketch one, you need a strategy for finding its key features. In this lesson you will choose among four strategies and use the result to describe the graph completely.

The key features of a quadratic function

A complete description includes:

the direction of opening (sign of a)
the vertex and the axis of symmetry
the maximum or minimum value
the y-intercept (c) and its mirror point
the zeros, if there are any
the domain {x ∈ ℝ} and the range

Four strategies for the vertex

Factor (when it factors easily): f(x) = a(x − r)(x − s). The axis is halfway between the zeros. See Sketching from Factored Form.
Complete the square: f(x) = a(x − h)² + k. See From Standard Form to Vertex Form: Completing the Square.
Partial factoring: factor x out of the first two terms only, f(x) = x(ax + b) + c. Then f(x) = c when x = 0 or when ax + b = 0, that is x = −b/a. These two points have the same height c, so the axis is halfway between them, at x = −b ÷ (2a).
Use h = −b ÷ (2a) directly (from From Vertex Form to Standard Form), then substitute to get k.

Strategies 3 and 4 work for every quadratic, even ones that do not factor.

A decision path for sketching from standard form. Whichever route you take, finish with the y-intercept, its mirror point, and the zeros if the graph has any.

Worked example

Example 1: It factors

Sketch f(x) = x² − 6x + 8 and list its key features.

Factor: two numbers with product 8 and sum −6 are −2 and −4. f(x) = (x − 2)(x − 4).
Zeros 2 and 4; axis x = 3; f(3) = (1)(−1) = −1, so vertex (3, −1).
a = 1 > 0: opens up, minimum −1. y-intercept 8, mirror point (6, 8).
D = {x ∈ ℝ}; R = {y ∈ ℝ | y ≥ −1}.

Zeros 2 and 4, vertex (3, −1), y-intercept 8, R = {y ∈ ℝ | y ≥ −1}.

Worked example

Example 2: Partial factoring

Sketch f(x) = 2x² + 8x + 3 and list its key features. Estimate the zeros, then find them with the quadratic formula.

It does not factor with integers. Partial factoring: f(x) = 2x(x + 4) + 3.
f(x) = 3 when 2x = 0 or x + 4 = 0, so at x = 0 and x = −4. Points (0, 3) and (−4, 3).
Axis halfway: x = (0 + (−4)) ÷ 2 = −2. f(−2) = 2(4) − 16 + 3 = −5. Vertex (−2, −5), a minimum.
Step pattern for a = 2: (−1, −3), (−3, −3), (0, 3) ✓.
Zeros: the curve crosses between −1 and 0, and between −4 and −3. Quadratic formula: x = (−8 ± √(64 − 24)) ÷ 4 = (−8 ± √40) ÷ 4 ≈ −0.42 and −3.58.
R = {y ∈ ℝ | y ≥ −5}.

Vertex (−2, −5), y-intercept 3, zeros ≈ −0.42 and −3.58, R = {y ∈ ℝ | y ≥ −5}.

Partial factoring: f(x) = 2x(x + 4) + 3 equals 3 at x = 0 and at x = −4. Those two points are mirror images, so the axis is halfway at x = −2 and the vertex is (−2, −5).

Worked example

Example 3: No zeros

Sketch f(x) = −x² + 4x − 7. Does it have zeros?

Use h = −b ÷ (2a) = −4 ÷ (2(−1)) = 2. f(2) = −4 + 8 − 7 = −3. Vertex (2, −3).
a = −1 < 0: opens down, so −3 is the maximum.
The highest point is below the x-axis and the graph opens down: no zeros.
y-intercept −7; mirror point (4, −7). More points: (1, −4), (3, −4).
R = {y ∈ ℝ | y ≤ −3}.

Vertex (2, −3), maximum −3, no zeros, y-intercept −7.

Worked example

Example 4: A decimal a

Sketch f(x) = 0.5x² − 3x + 2 and find its zeros to two decimal places.

h = −(−3) ÷ (2 × 0.5) = 3. f(3) = 4.5 − 9 + 2 = −2.5. Vertex (3, −2.5).
y-intercept 2; mirror point (6, 2).
Zeros: x = (3 ± √(9 − 4(0.5)(2))) ÷ (2 × 0.5) = 3 ± √5 ≈ 0.76 and 5.24.
Check symmetry: 0.76 and 5.24 are each 2.24 from 3 ✓.

Vertex (3, −2.5), y-intercept 2, zeros ≈ 0.76 and 5.24.

Just for fun

Partial factoring is for functions that won't fully open up to you. You can't get their zeros out of them right away, but you can still find two points with a lot in common.

Hands-on activity

Strategy relay

You need: graph paper, a pencil, a calculator, and a partner if possible. Desmos (free) is optional for checking.

Sketch each function with a different strategy (factoring, completing the square, partial factoring, h = −b ÷ (2a)):

(a) f(x) = x² − 2x − 3 (b) f(x) = x² + 4x + 1 (c) f(x) = −2x² + 4x + 1 (d) f(x) = 0.5x² + x − 4

For each one, list the vertex, axis, y-intercept, zeros (exact or to two decimal places) and range.
Swap with your partner and check each other's sketches using a different strategy, or check in Desmos.
Which strategy was fastest for each function? Why?

What to look for: (a) (x + 1)(x − 3): zeros −1, 3, vertex (1, −4). (b) vertex (−2, −3), zeros −2 ± √3 ≈ −0.27 and −3.73. (c) vertex (1, 3), opens down, zeros 1 ± √1.5 ≈ 2.22 and −0.22. (d) 0.5(x + 4)(x − 2): zeros −4 and 2, vertex (−1, −4.5). Factoring is fastest when it works; partial factoring or −b ÷ (2a) is quickest otherwise.

Your turn

Sketch f(x) = x² + 2x − 15. State the zeros, vertex and y-intercept.

Find the vertex of f(x) = x² − 4x + 7 and decide whether it has zeros.

Use partial factoring to find the vertex of f(x) = x² + 6x + 2.

Sketch f(x) = −2x² + 8x − 5. State the vertex, range and zeros (two decimal places).

Sketch f(x) = 3x² − 12x.

List all the key features of f(x) = −x² − 2x + 3.

More practice: warm-up

State the y-intercept of f(x) = 2x² − 5x + 9.

Does f(x) = −0.3x² + 5x + 1 open up or down?

Use h = −b ÷ (2a) to find the axis of f(x) = x² − 10x + 1.

f(x) = x² − 6x + 4. Besides x = 0, which x-value gives f(x) = 4?

The axis of f(x) = x² − 2x + 5 is x = 1. What is the vertex?

More practice: on your own

Sketch f(x) = 0.5x² − 3x + 2: vertex, y-intercept and zeros.

Sketch f(x) = 4x² + 4x − 3 by factoring.

Find the vertex and range of f(x) = −x² + 6x − 10. Does it have zeros?

Use partial factoring to find the vertex of f(x) = 2x² − 6x + 1.

A football punt's height is h(t) = −4.9t² + 19.6t + 0.5 metres. Find the maximum height and the time the ball hits the ground (two decimal places).

Challenge

Use partial factoring on f(x) = ax² + bx + c to prove that the axis of symmetry is x = −b ÷ (2a).

For which values of b does f(x) = x² + bx + 9 have its vertex on the x-axis?

Sketch f(x) = −0.25x² + 2x + 5 and list all key features.

Check your understanding

Quiz

What is the axis of symmetry of f(x) = 2x² − 12x + 1?

What is the vertex of f(x) = x² + 4x + 1?

Partial factoring of f(x) = x² − 8x + 5 gives two points with y = 5. They are…

How many zeros does f(x) = −x² + 2x − 3 have?

Which feature can you NOT read directly from standard form?

What is the range of f(x) = 3x² − 6x + 2?

For parents and teachers

Big idea: expectation A2.11 asks students to sketch the graph of a quadratic function given in standard form by using a suitable strategy, and to identify the key features of the graph.

How to help: ask your teen to name the strategy before they start ("This one factors" or "I'll use partial factoring"), and to finish every sketch with a list of key features. Checking the vertex in Desmos is quick and builds confidence.

Common difficulties:

Using h = b ÷ (2a) (missing the negative sign).
Forgetting to substitute h back in to find k.
In partial factoring, getting x = b/a instead of x = −b/a.
Giving a range without checking whether the parabola opens up or down.