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Core Connections Algebra 2, 2013 View details

1. Section 7.1

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Exercise 40 Page 326

The x-intercept is the point of intersection of the function with the x-axis. Similarly, the y-intercept is the point of intersection with the y-axis.

x-intercepts: -3±sqrt(6)/3
y-intercept: 1

Practice makes perfect

We want to find the x- and y-intercepts of a quadratic function. Let's begin with the x-intercept.

Finding the x-intercept

Think of the point where the graph of an equation crosses the x-axis. This is the x-intercept. The y-value of that ( x, y) coordinate pair is 0, so to find the x-intercept of the equation we should substitute 0 for y and solve for x.

y=3x^2+6x+1

Substitute

y= 0

0=3x^2+6x+1

We got a quadratic equation. To solve it we can use the Quadratic Formula.

ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a Let's identify the values of a, b, and c in our case. 0=3x^2+6x+1 ⇓ 3x^2+ 6x+ 1=0 We can see that a= 3, b= 6, and c= 1. Let's substitute these values into the Quadratic Formula.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- 6±sqrt(6^2-4( 3)( 1))/2( 3)

▼

Solve for x and Simplify

CalcPow

Calculate power

x=-6±sqrt(36-4(3)(1))/2(3)

Multiply

x=-6±sqrt(36-12)/6

SubTerm

Subtract term

x=-6±sqrt(24)/6

SplitIntoFactors

Split into factors

x=-6±sqrt(4*6)/6

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

x=-6±sqrt(4)*sqrt(6)/6

CalcRoot

Calculate root

x=-6±2*sqrt(6)/6

FactorOut

Factor out 2

x=2(-3±sqrt(6))/6

CancelCommonFac

Cancel out common factors

x=-3±sqrt(6)/3

The solutions for this equation are x= -3±sqrt(6)3. These are the x-intercepts.

Finding the y-intercept

Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the ( x, y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept.

y=3x^2+6x+1

Substitute

x= 0

y=3( 0)^2+6( 0)+1

ZeroPropMult

Zero Property of Multiplication

y=0+0+1

AddTerms

Add terms

y=1

A y-intercept of 1 means that the graph passes through the y-axis at the point (0, 1).