Exercise 14 Page 556

We have a quadratic function written in standard form. This kind of equation can give us a lot of information about the parabola by observing the values of a, b, and c. y=-4x^2-16x-3 ⇔ y=-4x^2+(- 16)x+(-3) We see that for the given equation, a=-4, b=-16, and c=-3.

x-value of the Vertex

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 - b2a and simplify.

- b/2a

SubstituteII

a= -4, b= -16

- -16/2(-4)

▼

Simplify

MultPosNeg

a(- b)=- a * b

- -16/-8

DivNegNeg

- a/- b=a/b

- 16/8

CalcQuot

Calculate quotient

-2

The axis of symmetry is the vertical line through the vertex and divides the 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=-2

y-value 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 the given value of a is negative, the parabola has a maximum value at the vertex. To find this value, think of y as a function of x, y=f(x). By substituting the x-value of the vertex into the given equation and simplifying, we will get the y-value of the vertex.

y=-4x^2-16x-3

Substitute

x= -2

y=-4( -2)^2-16( -2)-3

▼

Simplify right-hand side

CalcPow

Calculate power

y=-4(4)-16(-2)-3

MultNegPos

(- a)b = - ab

y=-16-16(-2)-3

MultNegNegOnePar

- a(- b)=a* b

y=-16+32-3

AddSubTerms

Add and subtract terms

y=13

The vertex is the point (- 2,13).