Question

fluency

1. the function ( y = f ( x ) ) is shown graphed below. answer the following questions based on the graph.

(a) evaluate each of the following:
( f ( - 2 ) = ) ( f ( 6 ) =)
(b) find the value of ( f ( 2 ) - f ( - 6 ) ). show how you
found your answer.
(c) state the values of all zeros of the function. circle
them on the graph.
(d) what is the value of the ( y )-intercept of the graph? (e) state the minimum and maximum values of the
function.
(f) state the coordinates of all turning points and label them as a maximum or a minimum point. (g) state an interval along which the function is only increasing.
(h) over the interval ( - 5 < x < 4 ), is the function positive or negative? how can you tell?

Explanation:

Step1: Evaluate \( f(-2) \)

To find \( f(-2) \), look at the \( y \)-value when \( x = -2 \) on the graph. The point at \( x=-2 \) has \( y = 8 \), so \( f(-2)=8 \).

Step2: Evaluate \( f(6) \)

For \( f(6) \), find the \( y \)-value at \( x = 6 \). The point at \( x = 6 \) has \( y = 2 \), so \( f(6)=2 \).

Step3: Evaluate \( f(2) \)

Locate \( x = 2 \) on the graph. The \( y \)-value at \( x = 2 \) is \( -2 \), so \( f(2)=-2 \).

Step4: Evaluate \( f(-6) \)

Find the \( y \)-value at \( x=-6 \). The point at \( x=-6 \) has \( y = 8 \), so \( f(-6)=8 \).

Step5: Calculate \( f(2)-f(-6) \)

Substitute the values: \( f(2)-f(-6)=-2 - 8=-10 \).

Step6: Find zeros of the function

Zeros are the \( x \)-values where \( y = 0 \). From the graph, the zeros are \( x=-4,x = 9 \).

Step7: Find \( y \)-intercept

The \( y \)-intercept is the \( y \)-value when \( x = 0 \). From the graph, \( y=-2 \), so the \( y \)-intercept is \( -2 \).

Step8: Find minimum and maximum

The maximum value is the highest \( y \)-value on the graph, which is \( 8 \). The minimum value is the lowest \( y \)-value, which is \( -4 \).

Step9: Find turning points

Turning points are where the function changes direction. The points are \( (-6,8) \) (maximum), \( (-3,-4) \) (minimum), \( (5,3) \) (maximum).

Step10: Find increasing interval

An increasing interval is where \( y \) increases as \( x \) increases. From the graph, the interval \( -3

Step11: Determine sign over \( -5 < x < 4 \)

For \( -5 < x < 4 \), look at the \( y \)-values. The \( y \)-values are negative (below the \( x \)-axis), so the function is negative.

Answer:

(a) \( f(-2)=8 \), \( f(6)=2 \)
(b) \( f(2)-f(-6)=-10 \)
(c) Zeros: \( x=-4,x = 9 \)
(d) \( y \)-intercept: \( -2 \)
(e) Minimum: \( -4 \), Maximum: \( 8 \)
(f) Turning points: \( (-6,8) \) (maximum), \( (-3,-4) \) (minimum), \( (5,3) \) (maximum)
(g) Increasing interval: \( -3(h) Negative; because \( y < 0 \) for \( -5 < x < 4 \) on the graph.