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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Graphing y = ax² + bx + c: Every Quadratic Makes a Parabola

1.2 1.3About 65 minutes
Learning goals
  • I can identify a quadratic relation from its equation.
  • I can graph y = ax² + bx + c from a table of values and check it with technology.
  • I can predict whether a parabola opens up or down from the sign of a.
  • I can find the y-intercept from the equation.

Words to know

standard form
y = ax² + bx + c, with a ≠ 0.
degree
The highest power of the variable in a polynomial. A quadratic has degree 2.
opens up / opens down
A parabola opens up when a > 0 (it has a lowest point) and down when a < 0 (it has a highest point).
symmetric
Having two halves that are mirror images of each other.
y-intercept
The y-coordinate where a graph crosses the y-axis; for y = ax² + bx + c it is c.

Every quadratic draws the same kind of picture

In the last lesson you found that quadratic relations have constant second differences. Now you will graph them. You will discover that every relation of the form y = ax² + bx + c with a ≠ 0 has a graph with the same basic shape: a parabola. Some are wide and some are narrow, some open up and some open down, and they sit in different places on the grid, but they are all parabolas.

The standard form of a quadratic relation

y = ax² + bx + c, where a, b and c are numbers and a ≠ 0.

  • The x² term is what makes it quadratic. The highest power of x is 2 (the degree is 2).
  • b and c may be 0. For example y = x² (b = 0, c = 0), y = x² − 4 (b = 0) and y = 3x² + 5x (c = 0) are all quadratic.
  • If a = 0, the x² term disappears and y = bx + c is linear. That is why a ≠ 0.

Examples of non-quadratic relations: y = 5x − 2 (linear), y = x³ (degree 3), y = 2ˣ (x is in the exponent).

Worked example

Example 1: Graph y = x² − 4 from a table

Graph y = x² − 4 for x from −3 to 3. Then check the graph with technology.

  1. Make a table. x = −3: 9 − 4 = 5. x = −2: 4 − 4 = 0. x = −1: 1 − 4 = −3. x = 0: 0 − 4 = −4. x = 1: −3. x = 2: 0. x = 3: 5.
  2. Points: (−3, 5), (−2, 0), (−1, −3), (0, −4), (1, −3), (2, 0), (3, 5).
  3. Plot the points and join them with a smooth curve. At the bottom, the curve rounds off at (0, −4); it does not come to a sharp point.
  4. Check: first differences −5, −3, −1, 1, 3, 5; second differences all 2.
  5. Technology check: type y = x^2 − 4 into Desmos or a graphing calculator. The parabola passes through exactly the same points.

A parabola opening up, with lowest point (0, −4), crossing the x-axis at −2 and 2. It is symmetric about the y-axis.

xy−4−3−2−101234−5−4−3−2−10123456(3, 5)(−3, 5)(0, −4)(2, 0)(−2, 0)y = x² − 4
The graph of y = x² − 4 from Example 1. The points on the left are mirror images of the points on the right: (−3, 5) matches (3, 5), and (−2, 0) matches (2, 0).
Worked example

Example 2: A parabola that opens down

Graph y = −x² + 2x + 3 for x from −2 to 4. Describe the graph.

  1. Be careful: −x² means −(x²). For x = −2: −(4) + 2(−2) + 3 = −4 − 4 + 3 = −5. Do not square the negative sign.
  2. The other values: x = −1: −1 − 2 + 3 = 0; x = 0: 3; x = 1: −1 + 2 + 3 = 4; x = 2: −4 + 4 + 3 = 3; x = 3: −9 + 6 + 3 = 0; x = 4: −16 + 8 + 3 = −5.
  3. First differences: 5, 3, 1, −1, −3, −5. Second differences: −2 each time (a = −1, so 2a = −2).
  4. Plot and join smoothly. The highest point is (1, 4). The graph is symmetric about the vertical line x = 1: (0, 3) matches (2, 3), and (−1, 0) matches (3, 0).

A parabola opening down, with highest point (1, 4), crossing the x-axis at −1 and 3, symmetric about x = 1.

xy−3−2−1012345−6−5−4−3−2−1012345(1, 4)y = −x² + 2x + 3axis ofsymmetry x = 1
The graph of y = −x² + 2x + 3 from Example 2. Because a = −1 is negative, the parabola opens down and has a highest point at (1, 4). The dashed line x = 1 cuts it into mirror-image halves.

The sign of a decides the direction

  • If a > 0, the parabola opens up (like a cup that holds water), and it has a lowest point.
  • If a < 0, it opens down (like an upside-down cup), and it has a highest point.

The second differences agree: they have the same sign as a.

xy−202−4−2024a > 0: opens upxy−202−4−2024a < 0: opens downpositive second differences → up; negative → down
Two parabolas that differ only in the sign of a. The one with a > 0 has a lowest point; the one with a < 0 has a highest point.
Worked example

Example 3: Investigating with technology

In Desmos, type y = ax² + bx + c and add sliders for a, b and c. Describe what happens as you change each one.

  1. Move a. Positive a: opens up. Negative a: opens down. The bigger the size of a (ignoring the sign), the narrower the parabola. When a reaches 0, the graph flattens into a straight line, which is why a ≠ 0.
  2. Move c. The whole parabola slides up or down, and the point where it crosses the y-axis is always (0, c). (At x = 0, y = a(0)² + b(0) + c = c.)
  3. Move b. The parabola slides sideways and up or down at the same time; its lowest or highest point moves along a curved path. The effect of b is harder to see, which is one reason you will soon learn a second form, y = a(x − h)² + k.

Every setting with a ≠ 0 gives a parabola. a controls direction and width; c is the y-intercept; b shifts the parabola in a less obvious way.

Just for fun

A funny-but-true fact: if you throw a ball in a perfect vacuum, its path is a perfect parabola. On Earth, air resistance squashes the far side a little, so a real baseball's path is only almost a parabola. Even nature rounds off its answers.

Hands-on activity

Parabolas in your kitchen

You need: a flashlight, a wall in a dark room, and optionally a garden hose or a drinking-water tap.

  1. Hold the flashlight close to the wall and point it straight at the wall: you see a circle. Slowly tilt it until the edge of the bright patch is no longer closed: one curved edge of the patch is close to a parabola.
  2. With a grown-up's permission, point a gentle stream of water from a hose (outside, in warm weather) or watch the water arc from a drinking fountain. Sketch the shape you see.
  3. Look for parabolas in pictures: arch bridges, the cables of a suspension bridge, the jet of a water fountain, a basketball shot.
  4. For one sketch, draw the vertical line that cuts it into two mirror halves, and mark its highest or lowest point.

What to look for: each sketch should be symmetric, with one highest (or lowest) point on the line of symmetry and smooth, rounded sides that get steeper as they go away from that point.

Your turn

1

Which of these are quadratic relations? (a) y = 3x² − 1 (b) y = 4x + 9 (c) y = −0.5x² + x (d) y = x³ + x²

2

Make a table of values for y = x² + 1 for x = −2 to 2.

3

Calculate y = −x² + 4 at x = −3. Explain the most common mistake.

4

Does y = −2x² + 5x − 1 open up or down? How do you know without graphing?

5

Where does y = 3x² − 7x + 6 cross the y-axis?

6

Make a table for y = x² − 2x for x = −1 to 3 and describe its graph.

More practice: warm-up

1

What is the name of the graph of a quadratic relation?

2

In y = ax² + bx + c, why must a not be 0?

3

Calculate 2x² at x = −3.

4

Does y = 0.1x² open up or down?

5

Give the values of a, b and c in y = 5x² − x + 8.

More practice: on your own

1

Graph y = 2x² − 3 using x = −2 to 2. Give the lowest point and the second differences.

2

Graph y = −x² − 2x for x = −3 to 1. Find its highest point and the line of symmetry.

3

A table for a parabola has the points (0, 7), (1, 2), (2, −1), (3, −2), (4, −1). Predict y at x = 5 and x = 6 using symmetry.

4

Write the equation of a quadratic relation that opens down and crosses the y-axis at (0, 10).

5

Rewrite y = (x + 1)(x − 3) in the form y = ax² + bx + c. Is it quadratic?

Challenge

1

Is y = (x + 2)² − x² a quadratic relation? Expand and decide.

2

The parabola y = x² + bx passes through (2, 10). Find b and the other x-intercept.

3

Explain why every parabola y = ax² + bx + c crosses the y-axis exactly once, but may cross the x-axis 0, 1 or 2 times.

Check your understanding

Quiz

1.

Which relation is quadratic?

2.

Which way does y = −0.5x² + 6x − 1 open?

3.

What is the y-intercept of y = 2x² − 5x + 9?

4.

For y = −x² + 1, what is y when x = 3?

5.

Why does the standard form require a ≠ 0?

6.

A parabola passes through (−1, 6) and is symmetric about the y-axis. Which other point must it pass through?

For parents and teachers

Big idea: Expectation 1.2 also asks students to see, with and without technology, that every relation y = ax² + bx + c (a ≠ 0) graphs as a parabola. Students should make tables by hand and then check them on a graphing tool. The direction of opening (sign of a) and the y-intercept (c) are the first features they read from the equation.

How to help: Have your teen graph one relation by hand on graph paper, then type it into Desmos (free, desmos.com) to check. Ask: "How did you know it would open down before you graphed it?"

Common difficulties:

  • Squaring the negative sign: −x² at x = 3 is −9, but (−3)² is +9. Have them write brackets when substituting: −(3)².
  • Pointy bottoms: the vertex of a parabola is rounded, not a V.
  • Joining points with a ruler. Parabolas are curves all the way along.
  • Calling y = (x + 2)² − x² quadratic without simplifying.