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Pearson Algebra 2 Common Core, 2011

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Pearson Algebra 2 Common Core, 2011 View details

3. Modeling With Quadratic Functions

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Exercise 24 Page 213

Consider the standard form of a quadratic function. Substitute the given points to get a system of equations and solve it to find the coefficients. Then, substitute any value of x and find its corresponding y-value.

Example Point: (2,2)

Practice makes perfect

Let's start by considering the standard form of a quadratic function. y=ax^2+bx+c Next, we substitute the three given points in the equation above to obtain three equations.

y=ax^2+bx+c
Point	Substitute	Simplify
( -1, 8)	8=a( -1)^2+b( -1)+c	a-b+c=8
( 0, 4)	4=a( 0)^2+b( 0)+c	c=4
( 1, 2)	2=a( 1)^2+b( 1)+c	a+b+c=2

We can use the three obtained equations to write a system of equations. a-b+c=8 & (I) c=4 & (II) a+b+c=2 & (III) From Equation (II), we already know that c=4. Let's substitute this value in Equation (I) and Equation (III), and simplify.

a-b+c=8 c=4 a+b+c=2

(I), (III): c= 4

a-b+ 4=8 c=4 a+b+ 4=2

(I), (III): LHS-4=RHS-4

a-b=4 c=4 a+b=- 2

We can now find the values of a and b by using the Elimination Method. To do so, let's add Equation (III) to Equation (I).

a-b=4 & (I) c=4 & (II) a+b=- 2 & (III)

(I): Add (III)

a-b+( a+b)=4+( - 2) c=4 a+b=- 2

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(I): Solve for a

(I): Remove parentheses

a-b+a+b=4+(- 2) c=4 a+b=- 2

(I): Add terms

2a=4+(- 2) c=4 a+b=- 2

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

2a=2 c=4 a+b=- 2

(I): .LHS /2.=.RHS /2.

a=1 c=4 a+b=- 2

Finally, we can find b by substituting 1 for a in Equation (III).

a=1 & (I) c=4 & (II) a+b=- 2 & (III)

(III): a= 1

a=1 c=4 1+b=- 2

(III):LHS-1=RHS-1

a= 1 c= 4 b= - 3

We can now write the equation of the corresponding parabola. y = 1x^2 +( - 3)x + 4 ⇕ y=x^2-3x+4 To find another point that also lies on the parabola, we can substitute any value of x and find its corresponding y-value. For example, let's substitute x=2.

y = x^2 - 3x + 4

x= 2

y = 2^2 - 3( 2) + 4

▼

Evaluate right-hand side

Calculate power

y = 4 - 3(2) + 4

Multiply

y = 4 - 6 + 4

Add and subtract terms

y = 2

Therefore, the point (2,2) also lies on the parabola. Remember, this is just an example point, since we can find infinitely many points.

Subchapter links

Lesson Check p.212

Practice and Problem-Solving Exercises p.212-214

Standardized Test Prep p.214

Mixed Review p.214