Year 9 AC9M9A04
Summary of important concepts Print or save as PDF
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. 
