Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
2. Characteristics of Quadratic Functions
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Exercise 42 Page 62

Start by identifying a, b, and c. The maximum value of the given quadratic function is f ( - b2a ).

Maximum Value: 8
Domain: All real numbers
Range: y ≤ 8
Increasing Interval: To the left of x=- 1
Decreasing Interval: To the right of x=- 1

Practice makes perfect

For the quadratic function f(x)=ax^2+bx+c, the y-coordinate of the vertex is the maximum value of the function when a<0.

Let's identify the values of a, b, and c in the given quadratic function. g(x)=- 3x^2-6x+5 ⇕ g(x)= - 3x^2+( - 6)x+ 5

We can see above that a= - 3, b= - 6, and c= 5. We will now use these values to find the desired information.

Maximum Value

Since a= - 3 is less than 0, the parabola will open downwards. This means it will have a maximum value, which is given by g ( - b2a ). Before we find the value of the function at this point, we need to substitute a= - 3 and b= - 6 in - b2a.
- b/2a
â–Ľ
Substitute values and evaluate
- - 6/2( - 3)
- - 6/- 6
- 6/6
- 1
Now we have to calculate g(- 1). To do so, we will substitute - 1 for x in the given function.
g(x)=- 3 x^2-6x+5
g(- 1)=- 3 (- 1)^2-6(- 1)+5
â–Ľ
Simplify right-hand side
g(- 1)=- 3(1)-6(- 1)+5
g(- 1)=-3+6+5
g(- 1)= 8
This tells us that the maximum value of the function is 8.

Domain and Range

Unless there are any specified restrictions on the x-values, the domain of a quadratic function is all real numbers. Therefore, the domain of this function is all real numbers. Furthermore, since a= - 3 is less than 0, the range is all values less than or equal to the maximum value, 8. Domain:& All real numbers Range:& y ≤ 8

Decreasing and Increasing Intervals

Since a= - 3 is less than 0, the function increases to the left of the maximum value and decreases to the right of the maximum value, which we know occurs at - 1. Increasing Interval:& To the left of - 1 Decreasing Interval:& To the right of - 1