Question

determine the absolute maximum and minimum of the given function. select all that apply
graph of a piecewise linear function on a grid with x-axis from 0 to 13 and y-axis from -300 to 300+
□ ( y = 400 )
□ ( y = -200 )
□ ( y = -300 )
□ ( y = 200 )
□ ( y = 300 )

Explanation:

Step1: Understand Absolute Max/Min

Absolute maximum is the highest \( y \)-value on the graph, absolute minimum is the lowest \( y \)-value.

Step2: Analyze the Graph

• The lowest point (minimum) has \( y = -300 \) (at \( x = 0 \)).
• The highest point (maximum) has \( y = 400 \) (at \( x = 7 \))? Wait, no, looking at the grid: the peak at \( x=7 \) is at \( y = 400 \)? Wait, the grid lines: each vertical grid? Wait, the \( y \)-axis: the bottom point is \( y=-300 \), the peak at \( x=7 \) is at \( y = 400 \)? Wait, no, let's check the options. The options are \( y=400 \), \( y=-200 \), \( y=-300 \), \( y=200 \), \( y=300 \).

Wait, the graph starts at \( (0, -300) \), goes up, then has peaks. The lowest point is \( y=-300 \) (absolute minimum). The highest peak: looking at the graph, at \( x=7 \), the point is at \( y = 400 \)? Wait, the grid: each square, let's see. The \( y \)-axis has 500, 300, 100, -100, -300. Wait, maybe the vertical grid lines: from \( y=-300 \) (bottom) to \( y=500 \) (top). The peak at \( x=7 \) is at \( y = 400 \)? Wait, no, maybe the labels: the first peak at \( x=5 \) is \( y=300 \), then at \( x=7 \) is higher, maybe \( y=400 \), and the lowest is \( y=-300 \) (at \( x=0 \)) and the other low at \( x=10 \) is \( y=-200 \), which is higher than -300. So absolute minimum is \( y=-300 \), absolute maximum is \( y=400 \)? Wait, but let's check the options. The options include \( y=400 \) and \( y=-300 \). Wait, maybe I misread. Wait, the graph: the starting point is \( (0, -300) \), then goes up, then at \( x=7 \), the point is at \( y = 400 \)? Wait, the grid lines: each horizontal line, the first below zero is -100, -300? Wait, no, the \( y \)-axis: the bottom mark is -300, then -100, 100, 300, 500. So the point at \( x=0 \) is \( (0, -300) \), the low at \( x=10 \) is \( (10, -200) \), which is higher than -300. The peak at \( x=7 \) is \( (7, 400) \)? Wait, the options have \( y=400 \) and \( y=-300 \). Let's confirm:

• Absolute minimum: the lowest \( y \)-value on the entire graph. The point at \( x=0 \) is \( y=-300 \), and the other low at \( x=10 \) is \( y=-200 \), so -300 is lower. So absolute minimum is \( y=-300 \).

• Absolute maximum: the highest \( y \)-value. The peak at \( x=7 \) is the highest, with \( y=400 \) (since it's above \( y=300 \) which is the peak at \( x=5 \)). So absolute maximum is \( y=400 \), absolute minimum is \( y=-300 \). Wait, but let's check the options. The options are:

\( \square y = 400 \)

\( \square y = -200 \)

\( \square y = -300 \)

\( \square y = 200 \)

\( \square y = 300 \)

So the absolute maximum is \( y=400 \), absolute minimum is \( y=-300 \). Wait, but maybe I made a mistake. Wait, the graph: let's count the grid squares. From \( y=-300 \) (bottom) to \( y=500 \) (top), each grid line is 100? So from -300, up 100 is -200, up 200 is -100, up 300 is 0, up 400 is 100? No, wait, maybe the \( y \)-axis labels: 500, 300, 100, -100, -300. So each major grid line is 200? Wait, 500 - 300 = 200, 300 - 100 = 200, etc. So the distance between 500 and 300 is 200, so each grid square is 100? Wait, no, the graph has a grid with squares. Let's see the point at \( x=3 \): \( y=100 \) (since it's on the 100 line). Then \( x=5 \): \( y=300 \) (on the 300 line). \( x=7 \): above 300, maybe 400 (since 300 + 100 = 400). \( x=10 \): \( y=-200 \) (below 0, above -300). \( x=0 \): \( y=-300 \) (bottom). So yes, absolute minimum is \( y=-300 \) (lowest), absolute maximum is \( y=400 \) (highest).

Answer:

The absolute maximum is \( y = 400 \) and the absolute minimum is \( y = -300 \). So the correct options are:

\( \boldsymbol{y = 400} \)

\( \boldsymbol{y = -300} \)