Grade by Grade
A. Quadratic Functions · Quadratic Expressions and Equations

Factors and x-Intercepts: Connecting Equations and Graphs

A1.5About 65 minutes
Learning goals
  • I can explain how the factors of a quadratic expression give the x-intercepts of its graph.
  • I can find the x-intercepts of a quadratic relation by factoring.
  • I can write the equation of a parabola from its x-intercepts and one other point.

Words to know

x-intercept
A point where a graph meets the x-axis; there y = 0.
zero (of a relation)
An x-value that makes y = 0.
factored form
y = a(x − r)(x − s), where r and s are the zeros.
repeated factor
A factor that appears twice, as in (x − 3)²; the graph touches the x-axis there.

Two languages, one parabola

You can describe a quadratic relation algebraically (an equation) or graphically (a parabola). Factoring connects the two. In this lesson you will investigate how the factors you use to solve a quadratic equation are related to the x-intercepts of the graph of the matching relation.

Hands-on activity

Investigation: factors and intercepts

You need: graph paper and a pencil, or a free online graphing tool such as Desmos or GeoGebra (on a computer or phone).

  1. Graph each relation: (a) y = x² − 2x − 8, (b) y = x² + 4x − 12, (c) y = 2x² + 5x − 3, (d) y = x² − 6x + 9. If you work by hand, make a table of values from x = −6 to x = 6.
  2. Record the x-intercepts of each graph.
  3. Factor each expression and solve the matching equation (for example, x² − 2x − 8 = 0).
  4. Compare your two lists. Write a rule that connects them.

What to look for: (a) (x − 4)(x + 2): intercepts 4 and −2. (b) (x + 6)(x − 2): −6 and 2. (c) (2x − 1)(x + 3): 0.5 and −3. (d) (x − 3)²: the graph only touches the axis at 3. Each factor gives one x-intercept: the x-value that makes that factor zero. The roots of the equation are exactly the x-intercepts of the graph.

What the investigation shows

  • A factor (x − r) makes y = 0 when x = r, so the graph crosses the x-axis at (r, 0).
  • A factor (px − q) gives the intercept x = q/p. For example, 2x − 1 = 0 when x = 1/2.
  • The roots of ax² + bx + c = 0 are the x-intercepts (zeros) of y = ax² + bx + c. Solving the equation and reading the graph are two ways of finding the same numbers.
  • The equation y = a(x − r)(x − s) is called factored form. It shows the zeros r and s directly.
−4−3−2−10123456−8−404812xyx + 2 = 0x − 4 = 0(0, −8)y = x² − 2x − 8= (x − 4)(x + 2)
Each factor of x² − 2x − 8 gives one x-intercept: x + 2 = 0 at x = −2 and x − 4 = 0 at x = 4. The axis of symmetry (dashed) is halfway between them, at x = 1.
Worked example

Example 1: From equation to intercepts

Find the x-intercepts and the y-intercept of y = 2x² + 5x − 3.

  1. Factor: ac = −6, sum 5: 6 and −1. 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (x + 3)(2x − 1).
  2. x-intercepts: set y = 0. x + 3 = 0 gives x = −3; 2x − 1 = 0 gives x = 0.5.
  3. y-intercept: set x = 0. y = 2(0) + 5(0) − 3 = −3.
  4. Check one intercept: 2(0.5)² + 5(0.5) − 3 = 0.5 + 2.5 − 3 = 0 ✓.

x-intercepts −3 and 0.5; y-intercept −3

The value of a

Multiplying by a number a does not change where a product is zero. So y = (x − 1)(x − 5), y = 2(x − 1)(x − 5) and y = −(x − 1)(x − 5) all have the same x-intercepts, 1 and 5. The value of a controls the shape: if a > 0 the parabola opens up, if a < 0 it opens down, and a larger |a| makes it narrower. You will explore a much more in the Quadratic Functions unit.

−101234567891011−8−4048xythick: y = (x − 1)(x − 5)dashed: y = 2(x − 1)(x − 5)thin: y = −(x − 1)(x − 5)
Three parabolas with the same zeros, x = 1 and x = 5. Changing a stretches or flips the graph but does not move the zeros.
Worked example

Example 2: From intercepts to equation

A parabola has x-intercepts −1 and 3 and passes through (1, −8). Find its equation in factored form and in standard form.

  1. The zeros give the factors (x + 1) and (x − 3), so y = a(x + 1)(x − 3).
  2. Substitute the point (1, −8): −8 = a(1 + 1)(1 − 3) = a(2)(−2) = −4a.
  3. So a = 2. Factored form: y = 2(x + 1)(x − 3).
  4. Standard form: 2(x² − 2x − 3) = 2x² − 4x − 6.

y = 2(x + 1)(x − 3), or y = 2x² − 4x − 6

−101234567−20246810xy(3, 0)y = (x − 3)²y = x² − 6x + 13
y = (x − 3)² has a repeated factor, so its graph touches the x-axis at one point, (3, 0). The dashed parabola y = x² − 6x + 13 never reaches the axis, and its expression does not factor.
Worked example

Example 3: One intercept, or none

How many x-intercepts do (a) y = x² − 6x + 9 and (b) y = x² − 6x + 13 have?

  1. (a) x² − 6x + 9 = (x − 3)². The only zero is x = 3, so the vertex sits on the x-axis: one x-intercept, (3, 0).
  2. (b) Product 13, sum −6: the pairs (1, 13) and (−1, −13) have sums ±14, so it does not factor. That alone does not prove there are no intercepts (some graphs cross at irrational values). But x² − 6x + 13 = (x − 3)² + 4, which is always at least 4, so y is never 0: no x-intercepts.
  3. The lesson Real Roots, x-Intercepts and the Discriminant gives a quick test for counting intercepts every time.

(a) one (b) none

Remember

Roots of the equation ax² + bx + c = 0 = zeros of the relation y = ax² + bx + c = x-intercepts of its graph. Three names, one set of numbers.

Just for fun

Why do parabolas make great friends?

They always tell you exactly where they cross the line.

Your turn

1

Find the x-intercepts of y = (x − 6)(x + 1).

2

Find the x-intercepts of y = x² + 4x − 12.

3

Find the x-intercepts of y = 3x² − 12.

4

Find the x-intercepts of y = −x² + 9x − 20. Does it open up or down?

5

Write the equation of the parabola with a = 1 and zeros 2 and 7, in standard form.

6

A parabola has zeros −3 and 1 and y-intercept 6. Find its equation.

7

Find the axis of symmetry of y = (x − 2)(x + 8).

More practice: warm-up

1

x-intercepts of y = (x − 5)(x − 9)?

2

x-intercepts of y = x(x + 4)?

3

x-intercepts of y = (2x − 3)(x + 1)?

4

How many x-intercepts does y = (x + 2)² have?

5

What is the y-intercept of y = (x − 3)(x + 5)?

More practice: on your own

1

Find the x-intercepts and axis of symmetry of y = 2x² − 7x + 3.

2

A parabola has zeros −4 and 2 and passes through (−2, 8). Find its equation in standard form.

3

How are y = (x − 1)(x − 5) and y = −3(x − 1)(x − 5) alike and different?

4

Explain why y = x² + 4 has no x-intercepts.

5

A ball's height is h = −5t² + 15t + 20 (metres, t in seconds). Factor to find when it lands.

Challenge

1

A parabola has x-intercepts −2 and 6, and its maximum value is 8. Find its equation.

2

Find a quadratic relation with x-intercepts 2/3 and −5 and y-intercept 10.

3

y = x² + bx + 12 has an x-intercept at x = 2. Find the other x-intercept and b.

Check your understanding

Quiz

1.

The x-intercepts of y = (x + 3)(x − 8) are:

2.

The roots of x² − x − 12 = 0 are 4 and −3. What does that tell you about y = x² − x − 12?

3.

Which relation has x-intercepts 0.5 and −4?

4.

y = (x − 1)(x − 7) and y = 5(x − 1)(x − 7) have:

5.

The graph of y = (x + 4)² meets the x-axis:

6.

A parabola has zeros 1 and 5. Its axis of symmetry is:

For parents and teachers

Big idea (A1.5): through investigation, students connect the factors used to solve a quadratic equation with the x-intercepts of the graph. Roots, zeros and x-intercepts are the same numbers.

How to help: if you have a phone or computer, open a free graphing tool together and type in y = (x − 2)(x + 5). Ask your teen to predict the x-intercepts before pressing enter. Then try changing the number in front.

Common difficulties:

  • Sign errors: the factor (x + 3) gives the intercept −3, not +3.
  • For a factor like (2x − 1), giving 1 instead of 1/2.
  • Thinking that a changes the zeros. It changes the shape and direction only.
  • Assuming every parabola has two x-intercepts. A repeated factor gives one; some have none.