  The graph of a quadratic function is called a parabola.

A quadratic graph is the graph of a quadratic function. This module describes the graphing of quadratic functions. A quadratic function has the form $$y=ax^{2}+bx+c$$ where $$a\neq0$$ .

The graph of a quadratic function is called a parabola.

To sketch a parabola, find and label:

1. the $$y$$-intercept (put $$x=0$$)

2. the $$x$$-intercepts (put $$y=0$$)

3. the vertex (turning point)

The co-ordinates of the vertex are given by:

$$x$$ co-ordinate $$\left(-\dfrac{b}{2a}\right)$$

$$y$$ co-ordinate: substitute the value of the $$x$$ co-ordinate in the equation for $$y$$.

A parabola is symmetrical about a vertical line through the vertex.

If $$a>0$$, then the parabola opens upwards (and has a minimum turning point). If $$a<0$$, then the parabola opens downwards (and has a maximum turning point). A quadratic function may also be written in turning point form: $$y=a(x-h)^{2}+k$$ , where $$(h,k)$$ is the turning point.

#### Examples

$$y=(x-3)^{2}+4$$ has a turning point at $$(3,4)$$

$$y=(x+5)^{2}+2$$ has a turning point at $$(-5,2)$$

$$y=2(x+1)^{2}$$ can be written as $$y=2(x+1)^{2}+0$$ and has a turning point at $$(-1,0)$$

$$y=x^{2}-7$$ can be written as $$y=(x-0)^{2}-7$$ and has a turning point at $$(0,-7)$$

$$y=6-(x-2)^{2}$$ can be written as $$y=-(x-2)^{2}+6$$ and has a turning point at $$(2,6)$$

### Sketching a Parabola

To sketch a parabola, find and label:

1. the $$y$$-intercept (put $$x=0$$)

2. the $$x$$-intercepts (put $$y=0$$)

3. the vertex (turning point)

#### Examples

1. Sketch $$y=x^{2}$$

Intercepts $$x=0$$ , $$y=0$$

Turning point $$(0,0)$$ 1. Sketch $$y=(x-1)^{2}-2$$

$$y$$-intercept: $$x=0$$ , $$y=-1$$

$$x$$-intercepts: $$y=0$$ , $$x=\pm\sqrt{2}+1$$

Turning point: $$(1,-2)$$ 1. Sketch $$y=x^{2}+3$$

$$y$$-intercept: $$x=0$$ , $$y=3$$

$$x$$-intercepts: $$y=0$$ , $$0=x^{2}+3\Rightarrow x^{2}=-3$$ no solution, no $$x$$-intercepts

Turning point: $$(0,3)$$ 1. Sketch $$y=4-2(x+3)^{2}$$

$$y$$-intercept: $$x=0$$ , $$y=-14$$

$$x$$-intercepts: $$y=0$$ ,

$$0=4-2(x+3)^{2}$$

$$\Rightarrow(x+3)^{2}=2$$

$$\Rightarrow x+3=\pm\sqrt{2}$$

$$\Rightarrow x=-3\pm\sqrt{2}$$

Turning point: $$(-3,4)$$ #### See Exercise 2

1. Sketch the graph $$y=x^{2}+2x-8$$

$$y$$-intercept: $$x=0$$ , $$y=-8$$

$$x$$-intercepts: $$y=0$$ ,

$$0=x^{2}+2x-8$$

$$\Rightarrow0=(x+4)(x-2)$$

$$\Rightarrow x=-4$$ or $$x=2$$

Turning point: This equation is not in turning point form so we use the equation for the $$x$$-coordinate of the turning point: $$x=\left(-\dfrac{b}{2a}\right)$$

In this example $$a=1$$ , $$b=2$$

therefore, the $$x$$-coordinate of the turning point is $$\left(-\dfrac{2}{2\times1}\right)=-1$$

Since $$y=x^{2}+2x-8$$ the $$y$$-coordinate of the turning point is $$y=(-1)^{2}+2(-1)-8=-9$$

$$T.P.=(-1,-9)$$ ### Exercise 1

State the turning point of the graphs of the following functions.

1. $$y=(x-1)^{2}+5$$

2. $$y=5(x-4)^{2}-12$$

3. $$y=(x+2)^{2}+3$$

4. $$y=-3(x+5)^{2}-3$$

5. $$y=(x-6)^{2}$$

6. $$y=-4x^{2}+3$$

1. $$(1,5)\ \ \$$ (b) $$(4,12)\ \ \$$ (c) $$(2,3)\ \ \$$ (d) $$(5,3)\ \ \$$ (e) $$(6,0)\ \ \$$ (f) $$(0,3)$$

### Exercise 2

Sketch graphs of the following.

1. $$y=x^{2}-7$$

2. $$y=(x-2)^{2}+1$$

3. $$y=4-(x+3)^{2}$$

4. $$y=(x-2)^{2}$$

5. $$y=-(x-1)^{2}-1$$     ### Exercise 3

Sketch the graphs of the following functions:

1. $$y=x^{2}-x-6$$

2. $$y=-x^{2}-2x+8$$

3. $$y=x^{2}-4x$$

4. $$y=-2x^{2}-6x$$

5. $$y=x^{2}-9$$     What's next... FG9 Graphs and transformations

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