Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Sketching y = ax² + bx + c: Choosing a Method

3.6 4.1About 75 minutes

Learning goals

I can compare the standard, factored and vertex forms and say what each one shows.
I can sketch a quadratic relation using intercepts and symmetry, or by completing the square and applying transformations.
I can use partial factoring to find the axis of symmetry and vertex.
I can use graphing technology to find zeros and a maximum or minimum, and interpret them.

Words to know

standard form
y = ax² + bx + c. The constant c is the y-intercept.

factored form
y = a(x − r)(x − s). The zeros are r and s.

vertex form
y = a(x − h)² + k. The vertex is (h, k).

partial factoring
Factoring only the x² and x terms, as in y = x(x − 6) + 2, to find two points with the same y-value.

step pattern
From the vertex of y = a(x − h)² + k, moving 1, 2, 3 units sideways moves a × 1, a × 4, a × 9 units up or down.

One parabola, three equations

The same parabola can be written in three ways. For example, the relation y = x² − 2x − 3 can be written as

standard form: y = x² − 2x − 3
factored form: y = (x − 3)(x + 1)
vertex form: y = (x − 1)² − 4

Expand the last two and you will get the first. Each form makes different features easy to see, so a smart grapher first asks: Which form do I have, and which form would help most?

Form	Equation	Shows directly	Good method for sketching
Standard	y = ax² + bx + c	y-intercept c; direction of opening (sign of a)	Factor it, complete the square, partial factor, or use technology
Factored	y = a(x − r)(x − s)	zeros r and s	Intercepts and symmetry: axis halfway between the zeros
Vertex	y = a(x − h)² + k	vertex (h, k); max or min value k	Transformations of y = x² and a step pattern

Choosing a sketching method based on the form of the equation.

Worked example

Example 1: Intercepts and symmetry

Sketch y = −x² + 4x + 5.

Factor: y = −(x² − 4x − 5) = −(x − 5)(x + 1).
Zeros: x = 5 and x = −1.
Axis: x = (5 + (−1)) ÷ 2 = 2.
Vertex: y = −(2 − 5)(2 + 1) = −(−3)(3) = 9. Vertex (2, 9), a maximum (a = −1).
y-intercept: c = 5. By symmetry, (4, 5) is also a point.
Plot (−1, 0), (5, 0), (2, 9), (0, 5) and (4, 5), and join them with a smooth curve that opens down.

A downward parabola with zeros −1 and 5 and maximum point (2, 9).

Worked example

Example 2: Complete the square and transform

Sketch y = 3x² − 12x + 1.

It does not factor nicely, so complete the square: y = 3(x² − 4x + 4 − 4) + 1 = 3(x − 2)² − 11.
Transformations of y = x²: vertical stretch by 3, translate 2 right and 11 down. Vertex (2, −11), opens up.
Step pattern: for y = x², from the vertex go 1 right and up 1, 2 right and up 4. Multiply by a = 3: up 3 and up 12.
Points: (3, −8) and (1, −8); (4, 1) and (0, 1). Check: the y-intercept of the standard form is 1 ✓.

Vertex (2, −11), axis x = 2, y-intercept 1, opening up and narrower than y = x².

y = 3(x − 2)² − 11 sketched from its vertex. One step right from the vertex goes up 3 (that is 3 × 1); two steps right go up 12 (3 × 4). Symmetry gives the points on the left.

Partial factoring: a shortcut to the axis

When a relation does not factor, you can still find the axis of symmetry quickly. Factor only the x² and x terms:

y = x² − 6x + 2 = x(x − 6) + 2.

When x = 0 or x = 6, the product x(x − 6) is zero, so y = 2 at both points. Two points with the same y-value are mirror images, so the axis is halfway between them: x = 3. Substitute: y = 9 − 18 + 2 = −7. The vertex is (3, −7).

Partial factoring for y = x² − 6x + 2. The points (0, 2) and (6, 2) are at the same height, so the axis of symmetry is halfway between them, at x = 3.

Worked example

Example 3: Using technology

A ball is kicked from 1.5 m above the ground. Its height in metres after t seconds is modelled by h = −4.9t² + 50t + 1.5. Use graphing technology to find the maximum height and when the ball lands.

Enter y = −4.9x² + 50x + 1.5 in a graphing tool (such as Desmos or a graphing calculator). Use a window from 0 to 11 for x and 0 to 140 for y.
Tap or trace to the maximum point: about (5.10, 129.1).
Find where the graph meets the horizontal axis (t > 0): about t = 10.23.
Interpret: the ball reaches about 129 m after about 5.1 s and lands after about 10.2 s. (That is a powerful kick! Models of real kicks use smaller numbers, but the method is the same.)

Maximum height ≈ 129.1 m at t ≈ 5.10 s; the ball lands at t ≈ 10.23 s.

The graph of h = −4.9t² + 50t + 1.5, drawn only for times from 0 until the ball lands. Graphing technology gives decimal values when the numbers are not friendly.

Which method should I use?

Factors easily? Use intercepts and symmetry.
Doesn't factor, but b ÷ 2a is a whole number? Complete the square, then transform y = x².
Only need the axis and vertex? Partial factoring is quick.
Messy decimals or a real-world model? Use graphing technology, then interpret the numbers in context.

Hands-on activity

Four methods, one sheet of paper

You need: graph paper (or a grid you draw), a pencil, and a graphing tool if a grown-up says you may use one.

Fold a sheet of graph paper into four sections.
Sketch one relation in each section, using a different method each time:

y = x² − 2x − 8 (intercepts and symmetry)
y = 2x² − 8x + 3 (complete the square)
y = x² + 4x + 7 (partial factoring)
y = −0.5x² + 3x + 2 (technology, or a table of values)

Label the vertex, axis and intercepts on each.
If you have a graphing tool, check each sketch.

Model answers: (1) zeros −2 and 4, vertex (1, −9). (2) y = 2(x − 2)² − 5, vertex (2, −5), y-intercept 3. (3) points (0, 7) and (−4, 7), axis x = −2, vertex (−2, 3), no zeros. (4) vertex (3, 6.5), y-intercept 2, zeros about −0.61 and 6.61.

Just for fun

Parabolas are the most polite guests at a dinner party. Whatever they do on the left side of the table, they do exactly the same on the right side. Their axis of symmetry is the salt shaker.

Your turn

Sketch y = x² − 4x − 12 using intercepts and symmetry. Give the zeros, vertex and y-intercept.

Sketch y = −x² + 6x − 5. Give the zeros, vertex and y-intercept.

Write y = 2x² + 8x + 3 in vertex form and describe the transformations of y = x².

Use partial factoring to find the vertex of y = x² + 4x + 7. Does it have zeros?

Which form of a quadratic relation shows each feature directly: (a) the zeros, (b) the vertex, (c) the y-intercept?

Use graphing technology to find the zeros (to two decimal places) and the vertex of y = 0.5x² − 3x + 1.

More practice: warm-up

State the y-intercept of y = 3x² − 5x + 4.

State the vertex of y = −2(x + 1)² + 6. Is it a max or a min?

State the zeros of y = (x − 3)(x + 8).

Does y = −0.5x² + x open up or down?

A parabola has zeros −4 and 10. What is its axis of symmetry?

More practice: on your own

Sketch y = 2x² − 4x − 6. Give the zeros, vertex and y-intercept.

Write y = −3x² + 12x − 7 in vertex form. State the vertex, the max or min value, and the y-intercept.

Use partial factoring to find the vertex of y = x² − 8x + 10.

A parabola has vertex (1, −4), a = 1, and passes through (3, 0). Write its equation in vertex, standard and factored form.

Use the step pattern to list four points (other than the vertex) on y = 2(x + 1)² − 3.

Challenge

For h = −4.9t² + 50t + 1.5 (Example 3), over what interval of time is the ball higher than 100 m? Use technology and give the times to one decimal place.

Use partial factoring on y = ax² + bx + c to show that the axis of symmetry is x = −b ÷ (2a). Test it on y = 3x² − 12x + 1.

A parabola opens down, has the same shape as y = 2x², and has zeros 1 and 5. Find its equation in all three forms.

Check your understanding

Quiz

Which form of a quadratic relation shows the zeros directly?

What is the vertex of y = 3(x − 2)² − 11?

Partial factoring writes y = x² − 6x + 2 as y = x(x − 6) + 2. What is the axis of symmetry?

For y = 2(x − 1)² + 3, if you move 1 unit right from the vertex, how far up do you move?

The height of a ball is h = −5t² + 20t + 1 (metres, seconds). What is the maximum height?

Which statement about y = x² + 4x + 7 is true?

For parents and teachers

Big idea: Expectation 3.6 asks students to sketch or graph y = ax² + bx + c using a variety of methods: intercepts and symmetry (e.g. y = x² − 2x − 8), completing the square and transformations (e.g. y = 3x² − 12x + 1), and technology (e.g. h = −4.9t² + 50t + 1.5). Expectation 4.1 adds finding zeros and the max or min from either a graph or an equation.

How to help: Ask "Which method did you choose, and why?" Encourage a labelled sketch every time: vertex, axis, intercepts. If you have a computer or phone, Desmos is free and lets your teen check a hand sketch instantly.

Common difficulties:

Forgetting to multiply the step pattern (1, 4, 9) by a.
Mixing up the sign of h in vertex form.
Using technology without adjusting the window, so the vertex or zeros are off the screen.