Math Problem Statement

solve

The graph of a function is shown on the coordinate plane. Which relationship represents a function with the same rate of change as the function graphed?

Solution

The question asks to determine which relationship represents a function with the same rate of change as the graphed function.

Step 1: Identify the slope (rate of change) from the graph.

The graph shows a straight line, indicating a linear function. To find the rate of change (slope), use the slope formula:

slope=ΔyΔx\text{slope} = \frac{\Delta y}{\Delta x}

From the graph, we can observe two points:

  • (0, 0) and (4, 3)

Now calculate the slope:

slope=3040=34\text{slope} = \frac{3 - 0}{4 - 0} = \frac{3}{4}

So, the rate of change (slope) for the graphed function is 34\frac{3}{4}.

Step 2: Examine each answer choice.

  • A: y=x+5y = x + 5
    The slope here is 11, since it's in the form y=mx+by = mx + b, where m=1m = 1. Not the correct choice because the slope is different.

  • B: Table with points (0, -5), (4, -12), (8, -19), (12, -26)
    Find the rate of change between points: slope=12(5)40=74\text{slope} = \frac{-12 - (-5)}{4 - 0} = \frac{-7}{4} This slope is 74-\frac{7}{4}.
    Not the correct choice because the slope is negative.

  • C: y=5x3y = 5x - 3
    The slope here is 55, since it's in the form y=mx+by = mx + b, where m=5m = 5. Not the correct choice because the slope is different.

  • D: Table with points (4, -6), (8, -11), (12, -16), (16, -21)
    Find the rate of change between points: slope=11(6)84=54\text{slope} = \frac{-11 - (-6)}{8 - 4} = \frac{-5}{4} This slope is 54-\frac{5}{4}, but upon checking all intervals, it's consistently the same.
    Not the correct choice because the slope is negative.

Conclusion:

None of the choices given directly match the slope 34\frac{3}{4} from the graph. There might be a mismatch between the slope of the function in the graph and the available options.

Would you like to go over more details or have further questions? Here are 5 relative questions that could expand the discussion:

  1. How can we derive the equation of a line given two points?
  2. What other ways can we calculate the rate of change from a graph?
  3. What is the significance of the slope in real-world applications?
  4. Can we determine the slope from just a table of values without graphing?
  5. How does a positive slope differ from a negative slope in interpreting functions?

Tip: Always look for two clear points on a graph to accurately determine the slope.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Slope-Intercept Form
Rate of Change

Formulas

Slope = (y2 - y1) / (x2 - x1)

Theorems

Slope of a line

Suitable Grade Level

Grades 7-9