Exercise 7 Page 434

We have a quadratic function written in standard form.

g (x) = a x^{2} + b x + c

This kind of equation can give us a lot of information about the parabola by observing the values of

a,

b,

and

c .

g (x) = 8 x^{2} - 8 x + 6 \Leftrightarrow g (x) = 8 x^{2} + (- 8) x + 6

We see that for the given equation

a = 8,

b = - 8,

and

c = 6 .

$x -$ coordinate of the Vertex

Consider the point at which the curve of the parabola changes direction.

This point is the vertex of the parabola, and defines the axis of symmetry. If we want to calculate the

x -

coordinate of this point, we can substitute the given values of

a

and

b

into the expression

- \frac{b}{2 a}

and simplify.

- \frac{b}{2 a}

SubstituteII

$a = 8$ , $b = - 8$

- \frac{- 8}{2 ( 8 )}

▼

Simplify

Multiply

- \frac{- 8}{1 6}

RemoveNegFracAndNum

$- \frac{- a}{b} = \frac{a}{b}$

\frac{8}{1 6}

ReduceFrac

$\frac{a}{b} = \frac{a / 8}{b / 8}$

\frac{1}{2}

$y -$ coordinate of the Vertex

The point at which the graph of a parabola changes direction also defines the maximum or minimum point of the graph. Whether the parabola has a minimum or maximum is determined by the value of $a .$

Since

a = 8

is positive, the parabola has a minimum value at the vertex. To find this value, substitute the

x -

coordinate of the vertex into the given function and simplify.

g (x) = 8 x^{2} - 8 x + 6

Substitute

$x = \frac{1}{2}$

g (\frac{1}{2}) = 8 (\frac{1}{2})^{2} - 8 (\frac{1}{2}) + 6

▼

Simplify right-hand side

PowQuot

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

g (\frac{1}{2}) = 8 (\frac{1}{4}) - 8 (\frac{1}{2}) + 6

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

g (\frac{1}{2}) = \frac{8}{4} - \frac{8}{2} + 6

CalcQuot

Calculate quotient

g (\frac{1}{2}) = 2 - 4 + 6

AddSubTerms

Add and subtract terms

g (\frac{1}{2}) = 4

The minimum value of the given function is

4 .

Extra

A Common Mistake

One common mistake when identifying the key features of a parabola algebraically is forgetting to include the negatives in the values of these constants. The standard form is addition only, so any subtraction must be treated as negative values of

a,

b,

or

c .

Let's look at an example.

a x^{2} + b x + c y = 2 x^{2} - 10 x + 13 \Leftrightarrow y = 2 x^{2} + (- 10 x) + 13

In this case, the values of

a,

b,

and

c

are

2,

- 10,

and

13 .

They are NOT

2,

10,

and

13 .

a = 2, b = 10, c = 13 \times a = 2, b = - 10, c = 13 ✓