Question

Use the method of symmetry to find the extreme value of each quadratic function and the value of x for which it occurs.

f(x)=(x-3)(x+8

a) x-intercepts are _________
b) midpoint of the x-intercepts is ___________
c) the extreme value is _______________
d) f(_) = ______________

Answer 1

`\sf a) \quad x = -8 \:\:and \:\:x = 3`

`\sf b) \quad \left(-(5)/(2),0\right)`

`\sf c) \quad \left(-(5)/(2),-(121)/(4)\right)`

`\sf d) \quad f(x)=\left(x+(5)/(2)\right)^2-(121)/(4)`

Explanation:

Part (a)

The x-intercepts are the x-values when the curve crosses the x-axis, so when the function is equal to zero:

`\implies f(x)=0`

`\implies (x-3)(x+8)=0`

Therefore:

`(x-3)=0 \implies x=3`

`(x+8)=0 \implies x=-8`

Therefore, the x-intercepts are x = -8 and x = 3

Part (b)

`\textsf{Midpoint}=\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)\quad \textsf{where}\:(x_1,y_1)\:\textsf{and}\:(x_2,y_2)\:\textsf{are the endpoints}}\right)`

`\sf \implies Midpoint=\left((-8+3)/(2),(0+0)/(2)\right)=\left(-(5)/(2),0\right)`

Therefore, the midpoint of the x-intercepts is (-5/2, 0)

Part (c)

The extreme value is the vertex. The vertex is:

• the minimum point if the parabola opens upwards
• the maximum point if the parabola open downwards

The x-coordinate of the vertex is the x-value of the midpoint of the x-intercepts. Therefore, the x-value of the extreme value is x = -5/2.

To find the y-coordinate, substitute this into the given equation:

`\implies \sf y=\left(-(5)/(2)-3\right)\left(-(5)/(2)+8\right)=-(121)/(4)`

Therefore, the extreme value is:

`\sf \left(-(5)/(2),-(121)/(4)\right)`

Part (d)

The function in vertex form is:

`\sf f(x)=\left(x+(5)/(2)\right)^2-(121)/(4)`

`\begin{aligned}\implies \sf f\left(-(5)/(2)\right)& = \sf \left(-(5)/(2)+(5)/(2)\right)^2-(121)/(4)\\\\ & = \sf 0 -(121)/(4)\\\\ & = \sf -(121)/(4) \end{aligned}`

Answer 2

• See below

Explanation:

The x-intercepts are the points with y- coordinate of 0

• (x - 3)(x + 8) = 0
• x - 3 = 0 or x + 8 = 0
• x = 3 or x = - 8

a) x-intercepts are

• 3 and - 8

b) midpoint of x-intercepts is

• (3 - 8)/2 = - 5/2 = - 2.5

c) The extheme value has x- coordinate of - 2.5 and y- coordinate

• (-2.5 - 3)(- 2.5 + 8) = (-5.5)(5.5) = - 30.25

d) f(- 2.5) = - 30.25