Cup up or cup down?

The idea

The graph of y = ax² + bx + c, a ≠ 0, is a curve called a parabola, which opens upwards if a > 0 and downwards if a < 0.

Which coefficient?

The graph of y = ax² + bx + c is a curve like a cup. It can open up or open down.

Which coefficient decides the direction?

• a
• b
• c

Flip the cup

Change a, b and c one at a time. What flips the cup?

Parabola

The graph of y = ax² + bx + c, a ≠ 0, is a curve called a parabola. It opens upwards if a > 0 and downwards if a < 0.

Only the sign of a decides the opening. Changing b or c moves the curve, but never flips it.

If a = 0, there is no x² term, so the graph is a straight line, not a parabola.

Notes

The graph of y = ax² + bx + c (a ≠ 0) is a parabola: upwards if a > 0, downwards if a < 0.

Check yourself

y = 3x² − 5x + 2 opens…

y = 4 − x² opens…

y = −2x² + 8x + 100 has a big positive b and c. Which way does it open?

What happens to y = ax² + bx + c when a = 0?

• upwards — correct. Yes! a = 3, which is greater than 0.
• downwards. The −5 is b. The sign of a decides, and a = 3 is positive.
• neither, it is a line. It has an x² term with a = 3, so it is a parabola.
• downwards — correct. Yes! The coefficient of x² is −1, so a < 0.
• upwards. The 4 is c. The coefficient of x² is −1, which is negative.
• downwards — correct. Yes! a = −2 < 0. b and c do not decide.
• upwards. b and c are positive, but only a decides. a = −2 is negative.
• It becomes a straight line — correct. Yes! Only bx + c is left, and its graph is a line.
• It opens upwards. With a = 0 there is no x² term, so there is no cup.
• It opens downwards. With a = 0 there is no x² term, so there is no cup.