Exercise 17 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. (0,3),(1,10),(2,19) 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 (0,3).

y=ax^2+bx+c

SubstituteII

x= 0, y= 3

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

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Simplify

CalcPow

Calculate power

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

ZeroPropMult

Zero Property of Multiplication

3=0+0+c

IdPropAdd

Identity Property of Addition

3=c

RearrangeEqn

Rearrange equation

c=3

We already found that c=3! y=ax^2+bx+3 Now we can use (1,10) to write our second equation.

y=ax^2+bx+3

SubstituteII

x= 1, y= 10

10=a( 1)^2+b( 1)+3

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Simplify

BaseOne

1^a=1

10=a(1)+b(1)+3

IdPropMult

Identity Property of Multiplication

10=a+b+3

SubEqn

LHS-3=RHS-3

7=a+b

RearrangeEqn

Rearrange equation

a+b=7

To find another equation, we will use (2,19).

y=ax^2+bx+3

SubstituteII

x= 2, y= 19

19=a( 2)^2+b( 2)+3

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Simplify

CalcPow

Calculate power

19=a(4)+b(2)+3

Multiply

Multiply

19=4a+2b+3

SubEqn

LHS-3=RHS-3

16=4a+2b

DivEqn

.LHS /2.=.RHS /2.

8=2a+b

RearrangeEqn

Rearrange equation

2a+b=8

We now have a system of two equations. a+b=7 & (I) 2a+b=8 & (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=7 2a+b=8

SysEqnSub

(II): Subtract (I)

a+b=7 2a+b-( a+b)=8- 7

Distr

(II): Distribute - 1

a+b=7 2a+b-a-b=8-7

SubTerms

(II): Subtract terms

a+b=7 a=1

We found that a=1. y=1x^2+bx+3 ⇔ y=x^2+bx+3 Let's now substitute 1 for a in Equation (I).

a+b=7 a=1

Substitute

(I): a= 1

1+b=7 a=1

SubEqn

(I): LHS-1=RHS-1

b=6 a=1

Now that we have all three values, we can complete the standard form equation of the parabola that passes through the given points. y=1x^2+6x+3 ⇔ y=x^2+6x+3 To help visualize this graph, we have plotted the given points and sketched the curve below.