grade 9 Graphs of Quadratic Functions

Graphs of Quadratic Functions

The graph of a quadratic function y = ax² + bx + c is a smooth, symmetric curve called a parabola. The picture uses the a, p, q sliders; the thrown-ball path lives in the previous application chapter. The sign of a decides whether the curve opens upward or downward, and a larger absolute value makes it narrower. Move a and the vertex p, q and the parabola widens, flips, and glides.

y = ax² — Role of a

value a1

📊 What a Determines

①a > 0: opens upward

②a < 0: opens downward

③Larger |a| → narrower; smaller |a| → wider

④a = 0 isn't a quadratic

y = a(x−p)²+q — Vertex Form

p (shift x)2

q (shift y)-1

🎯 Idea of Translation

①y=ax² → y=a(x−p)²: shifts +p along x-axis

②y=ax² → y=ax²+q: shifts +q along y-axis

③Vertex: (p, q), axis: x = p

④Watch signs: y=a(x−2)²+3 → vertex (2, 3). Defaults a=1, p=2, q=−1 give standard form y=x²−4x+3

Standard → Vertex Form

Standard Form

y = ax² + bx + c

Expanded form — coefficients direct

Vertex Form Conversion

y = a(x + b2a)² − b²−4ac4a

Complete the square to expose the vertex

🔄 Conversion Steps

①Factor a out of x²+bx in ax²+bx+c

②Add and subtract (b/2)²

③Group into a perfect square

④Vertex = (−b/2a, −(b²−4ac)/4a)

Putting the Properties Together

Reading the Graph

Vertex (p,q), axis: x=p, y-intercept: c, direction: sign of a

All info from a quadratic graph

💡 Graph Properties

①Axis of symmetry: x = p (x of vertex)

②y-intercept: set x=0, y=c

③x-intercepts: set y=0 → roots of the quadratic

④Max/min: y of the vertex

⑤Symmetry: f(p+t)=f(p−t)

Work It Out

Example 1

Find the coordinates of the vertex of y=(x−2)²+3.

In vertex form y=a(x−p)²+q, the vertex is (p, q).

y = (x−2)²+3

Here p=2, q=3, so the vertex is (2, 3).

vertex (2, 3)

▸ (2, 3)

(x−2) is the x−p form, so p=+2. Mind the sign.

Example 2

Rewrite y=x²−4x+1 in vertex form and find the vertex.

Complete the square on x²−4x: x²−4x = (x−2)²−4.

y = (x−2)²−4+1

So y=(x−2)²−3, and the vertex is (2, −3).

y = (x−2)²−3 ⇒ vertex (2, −3)

▸ y=(x−2)²−3, vertex (2, −3)

Add and subtract the square of half the x-coefficient: (−2)²=4.

Exam Wrap-up

Core Form

y=a(x−p)²+q — vertex (p,q), axis x=p

Starting point for every quadratic-function problem

G9 school-exam type

What is the axis of symmetry of y=x²−6x+5?

①x=−3

②x=−6

③x=3

④x=5

⑤x=6

▸ ③ x=3

The axis is x = −b/(2a) with a=1, b=−6.

x = −(−6)/(2·1)

Compute: x=3.

x = 6/2 = 3

🎯 Exam Points

①Sign of a → opens up/down

②Larger |a| → narrower

③Watch signs in y=a(x−p)²+q : (x−2) means p = +2

④Practice converting standard ↔ vertex form

⑤Use symmetry: f(p+t) = f(p−t)

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Applications of Quadratic Equations

Applications of Quadratic Functions

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