Interpreting Quadratic Functions in Standard Form

a

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 $c .$ $g (x) = - 9 x^{2} - 18 x - 1 \Leftrightarrow g (x) = - 9 x^{2} + (- 18) x + (- 1)$ We see that for the given equation $a = - 9,$ $b = - 18,$ and $c = - 1 .$ These values will give us information about the parabola. 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 -

value 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 = - 9

,

b = - 18

- \frac{- 1 8}{2 ( - 9 )}

Simplify

MultPosNeg

a (- b) = - a \cdot b

- \frac{- 1 8}{- 1 8}

DivNegNeg

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

- \frac{1 8}{1 8}

QuotOne

\frac{a}{a} = 1

- 1

Remember that the axis of symmetry is the vertical line that passes through the vertex, dividing a parabola into two mirror images. Since every point on this line will have the same $x -$ coordinate as the vertex, we can form its equation. $x = - 1$

b

We already know that the

x -

coordinate of the vertex is

- 1 .

To find its

y -

coordinate, we will substitute

- 1

by

x

in the formula.

g (x) = - 9 x^{2} - 18 x - 1

Substitute

x = - 1

g (- 1) = - 9 (- 1)^{2} - 18 (- 1) - 1

Simplify right-hand side

CalcPowCalculate power

g (- 1) = - 9 (1) - 18 (- 1) - 1

MultByOne

a \cdot 1 = a

g (- 1) = - 9 - 18 (- 1) - 1

MultNegNegOnePar

- a (- b) = a \cdot b

g (- 1) = - 9 + 18 - 1

AddSubTermsAdd and subtract terms

g (- 1) = 8

The vertex of the parabola is the point

(- 1, 8) .

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. $a x^{2} + b x + c y = - 9 x^{2} - 18 x - 1 \Leftrightarrow y = - 9 x^{2} + (- 18 x) + (- 1)$ In this case, the values of $a,$ $b,$ and $c$ are $- 9,$ $- 18,$ and $- 1 .$ They are NOT $- 9,$ $18,$ and $1 .$ $a = - 9, b = 18, c = 1 \times a = - 9, b = - 18, c = - 1 ✓$

Interpreting Quadratic Functions in Standard Form 1.19 - Solution

Extra