Exercise 18 Page 224

Before we try to write the equation of a specific parabola that passes through some given points, let's make sure we have enough points. We will need at least three. (0,0),(1,- 5),(2,0) To use the given points, we need to substitute their ( x, y) coordinate pairs into the standard form of a quadratic equation. y=a x^2+b x+c, a≠ 0 Doing so will create a system of equations that we can solve for the values of a, b, and c. Let's start with (0,0).

y=ax^2+bx+c

SubstituteII

x= 0, y= 0

0=a( 0)^2+b( 0)+c

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Simplify

CalcPow

Calculate power

0=a(0)+b(0)+c

ZeroPropMult

Zero Property of Multiplication

0=0+0+c

IdPropAdd

Identity Property of Addition

0=c

RearrangeEqn

Rearrange equation

c=0

We already found that c=0. y=ax^2+bx+0 ⇔ y=ax^2 +bx Now we can use (1,- 5) to write our second equation.

y=ax^2+bx

SubstituteII

x= 1, y= - 5

- 5=a( 1)^2+b( 1)

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Simplify

BaseOne

1^a=1

- 5=a(1)+b(1)

IdPropMult

Identity Property of Multiplication

- 5=a+b

RearrangeEqn

Rearrange equation

a+b=- 5

To find another equation, we will use (2,0).

y=ax^2+bx

SubstituteII

x= 2, y= 0

0=a( 2)^2+b( 2)

▼

Simplify

CalcPow

Calculate power

0=a(4)+b(2)

Multiply

Multiply

0=4a+2b

DivEqn

.LHS /2.=.RHS /2.

0=2a+b

RearrangeEqn

Rearrange equation

2a+b=0

We now have a system of two equations. a+b=- 5 & (I) 2a+b=0 & (II) Let's solve this system using the Elimination Method. We will start by subtracting Equation (I) from Equation (II) to eliminate the b-variable.

a+b=- 5 2a+b=0

SysEqnSub

(II): Subtract (I)

a+b=- 5 2a+b-( a+b)=0-( - 5)

SubNeg

(II): a-(- b)=a+b

a+b=- 5 2a+b-(a+b)=0+5

IdPropAdd

(II): Identity Property of Addition

a+b=- 5 2a+b-(a+b)=5

Distr

(II): Distribute - 1

a+b=- 5 2a+b-a-b=5

SubTerms

(II): Subtract terms

a+b=- 5 a=5

We found that a=5. y=5x^2+bx Let's now substitute 5 for a in Equation (I).

a+b=- 5 a=5

Substitute

(I): a= - 5

5+b=- 5 a=5

SubEqn

(I): LHS-5=RHS-5

b=- 10 a=5

Now that we have all three values, we can complete the standard form equation of the parabola that passes through the given points. y=5x^2+(- 10)x ⇕ y = 5x^2-10x To help visualize this graph, we have plotted the given points and sketched the curve below.