Question

Express each of the following quadratic functions in the form of f(x) = a (x - h)²+ k.Then,state the minimum or maximum value,axis of symmetry and minimum or maximum point. (a) f(x) = -2x² + 7x + 4.
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Answer 1

Given:

The function is:

$f(x)=-2x^2+7x+4$

To find:

Express the quadratic equation in the form of $f(x)=a(x-h)^2+k$, then state the minimum or maximum value,axis of symmetry and minimum or maximum point.

Solution:

The vertex form of a quadratic function is:

$f(x)=a(x-h)^2+k$              ...(i)

Where, a is a constant, (h,k) is the vertex and x=h is the axis of symmetry.

We have,

$f(x)=-2x^2+7x+4$

It can be written as:

$f(x)=-2\left(x^2-3.5x\right)+4$

Adding and subtracting square of half of coefficient of x inside the parenthesis, we get

$f(x)=-2\left(x^2-3.5x+(\dfrac{3.5}{2})^2-(\dfrac{3.5}{2})^2\right)+4$

$f(x)=-2\left(x^2-3.5x+(1.75)^2\right)-2\left(-(1.75)^2\right)+4$

$f(x)=-2\left(x-1.75\right)^2+2(3.0625)+4$

$f(x)=-2\left(x-1.75\right)^2+6.125+4$

$f(x)=-2\left(x-1.75\right)^2+10.125$                ...(ii)

On comparing (i) and (ii), we get

$a=-2,h=1.75,k=10.125$

Here, a is negative, the given function represents a downward parabola and its vertex is the point of maxima.

Maximum value = 10.125

Axis of symmetry : $x=1.75$

Maximum point = (1.75,10.125)

Therefore, the vertex form of the given function is $f(x)=-2\left(x-1.75\right)^2+10.125$, the maximum value is 10.125, the axis of symmetry is $x=1.75$ and the maximum point is (1.75,10.125).