Exercise 10 Page 556

To find the equation of the axis of symmetry and coordinates of the vertex of the graph of given quadratic function written in standard form, we must start by identifying the values of a, b, and c. y=x^2-8x-7 ⇔ y= 1x^2+( - 8)x+( - 7) We can see that a= 1, b= - 8, and c= - 7.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation x= - b2 a. Since we already know the values of a and b, we can substitute them into the formula.

x=- b/2a

SubstituteII

a= 1, b= - 8

x=- ( - 8)/2( 1)

▼

Simplify right-hand side

IdPropMult

Identity Property of Multiplication

x=-(- 8)/2

NegNeg

- (- a)=a

x=8/2

CalcQuot

Calculate quotient

x=4

The axis of symmetry of the parabola is the vertical line with equation x=4.

Calculating the Vertex

To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b. Vertex: ( - b/2 a, f(- b/2 a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=4. Therefore, the x-coordinate of the vertex is also 4. To find the y-coordinate, we need to substitute 4 for x in the given equation.

y=x^2-8x-7

Substitute

x= 4

y= 4^2-8( 4)-7

▼

Simplify right-hand side

CalcPow

Calculate power

y=16-8(4)-7

Multiply

Multiply

y=16-32-7

AddSubTerms

Add and subtract terms

y=- 23

We found the y-coordinate, and now we know that the vertex is (4,- 23).