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Core Connections Algebra 2, 2013 View details

1. Section 2.1

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Exercise 73 Page 82

Practice makes perfect

a At a function's x-intercept, the value of y is 0. Therefore, we can find the function's x-intercept(s) by setting y equal to 0 and solving for x.

y=2x^2+3x-5

Substitute

y= 0

0=2x^2+3x-5

RearrangeEqn

Rearrange equation

2x^2+3x-5=0

▼

Solve using the quadratic formula

UseQuadForm

Use the Quadratic Formula: a = 2, b= 3, c= - 5

x=- 3±sqrt(3^2-4( 2)( - 5))/2( 2)

CalcPowProd

Calculate power and product

x=- 3± sqrt(9+40)/4

AddTerms

Add terms

x=- 3± sqrt(49)/4

CalcRoot

Calculate root

x=- 3± 7/4

StateSol

State solutions

lcx=.(-3-7) /4. & (I) x=.(-3+7) /4. & (II)

(I), (II): Add and subtract terms

lx=.-10 /4. x=.4 /4.

(I), (II): Calculate quotient

lx_1=- 2.5 x_2=1

We have two x-intercepts at (1,0) and (- 2.5,0). To find the y-intercept, we have to substitute x with 0 and simplify.

y=2x^2+3x-5

Substitute

x= 0

y=2( 0)^2+3( 0)-5

▼

Evaluate right-hand side

CalcPow

Calculate power

y=2(0)+3(0)-5

ZeroPropMult

Zero Property of Multiplication

y=-5

Let's summarize the function's intercepts. x-intercepts:& (1,0) and (- 2.5,0) y-intercept:& (0,- 5)

b Like in Part A, we have to set y equal to 0 in order to find the value of x.

y=sqrt(2x-4)

Substitute

y= 0

0=sqrt(2x-4)

▼

Solve for x

RaiseEqn

LHS^2=RHS^2

0=2x-4

AddEqn

LHS+4=RHS+4

4=2x

DivEqn

.LHS /2.=.RHS /2.

2=x

RearrangeEqn

Rearrange equation

x=2

When we solve radical equations, it can produce extraneous solutions. Therefore, the solution must be verified by substituting it in the original equation.

0=sqrt(2x-4)

Substitute

x= 2

0? =sqrt(2( 2)-4)

▼

Evaluate right-hand side

Multiply

0? =sqrt(4-4)

SubTerm

Subtract term

0? =sqrt(0)

CalcRoot

Calculate root

0=0 ✓

Since x=2 makes a true statement it is a solution to the radical equation. To find the y-intercept, we set x equal to 0 and evaluate the right-hand side.

y=sqrt(2x-4)

Substitute

x= 0

y=sqrt(2( 0)-4)

▼

Evaluate right-hand side

ZeroPropMult

Zero Property of Multiplication

y=sqrt(0-4)

SubTerm

Subtract term

y=sqrt(-4)

Notice that when trying to find the y-intercept we ended up with a forbidden calculation, namely trying to calculate a negative square root. This means the function does not have a y-intercept. Let's summarize. x-intercepts:& (2,0) y-intercept:& Noy-intercept

Subchapter links

2.1.1 Modeling Non-Linear Data p.58-59

2.1.2 Parabola Investigation p.63-65

2.1.3 Graphing a Parabola Without a Table p.69-70

2.1.4 Rewriting in Graphing Form p.76-78

2.1.5 Mathematical Modeling with Parabolas p.81-82

Exercise 73 Page 82

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