Year 9 AC9M9A04
Graphing Quadratic Functions Identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically.
A quadratic makes a parabola. Three features fix the whole curve: where it crosses, which way it opens, and where it turns.
Method The shape is set by the sign of $a$
y = a x 2 + b x + c y = ax^2 + bx + c y = a x 2 + b x + c Positive $a$ opens upward. Negative $a$ opens downward.
The $y$-intercept is $c$
Set $x = 0$ and everything else vanishes.
The $x$-intercepts are the roots
Where the curve cuts the $x$-axis.
The turning point sits midway between the roots
Symmetry. The axis of symmetry runs through it.
Worked example e.g. Sketch y = x 2 − 4 \text{Sketch } y = x^2 - 4 Sketch y = x 2 − 4 Check the sign of a a a a = 1 > 0 a = 1 > 0 a = 1 > 0 Find the y y y x = 0 x = 0 x = 0 y = − 4 y = -4 y = − 4 Find the x x x y = 0 y = 0 y = 0 x 2 − 4 = 0 ⇒ x = ± 2 x^2 - 4 = 0 \ \Rightarrow\ x = \pm 2 x 2 − 4 = 0 ⇒ x = ± 2 The turning point sits midway between the roots. x = 0 ⇒ ( 0 , − 4 ) x = 0 \ \Rightarrow\ (0, -4) x = 0 ⇒ ( 0 , − 4 ) Plot the three points and draw a smooth curve. The parabola y equals x squared minus four with its intercepts marked x y y = x² − 4 Check the symmetry. Both roots are the same distance from the axis. ± 2 ✓ \pm 2 \quad\checkmark ± 2 ✓ Practice Find the intercepts first. The turning point follows from symmetry.
Reveal all answers 1
Which way does y = − x 2 + 3 open? \text{Which way does } y = -x^2 + 3 \text{ open?} Which way does y = − x 2 + 3 open? Answer Answer Downward, since a < 0 \text{Downward, since } a < 0 Downward, since a < 0 2
Find the y -intercept of y = x 2 + 5 x − 6 \text{Find the } y\text{-intercept of } y = x^2 + 5x - 6 Find the y -intercept of y = x 2 + 5 x − 6 Answer 3
Find the x -intercepts of y = x 2 − 9 \text{Find the } x\text{-intercepts of } y = x^2 - 9 Find the x -intercepts of y = x 2 − 9 Answer 4
Roots are 1 and 7. Find the axis of symmetry. \text{Roots are } 1 \text{ and } 7. \text{ Find the axis of symmetry.} Roots are 1 and 7. Find the axis of symmetry. Answer 5
Does y = x 2 + 1 cut the x -axis? \text{Does } y = x^2 + 1 \text{ cut the } x\text{-axis?} Does y = x 2 + 1 cut the x -axis? Answer Answer No. Its lowest point is ( 0 , 1 ) . \text{No. Its lowest point is } (0,1). No. Its lowest point is ( 0 , 1 ) . 6
How many x -intercepts can a parabola have? \text{How many } x\text{-intercepts can a parabola have?} How many x -intercepts can a parabola have? Answer Answer Two, one or none. \text{Two, one or none.} Two, one or none.