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Common Core Standards: Math

Grade 8

Functions 8.F.B.4

4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Some hills are hills, and some hills are mountains. Of course, some hills are alive with the sound of music, but that's a different story.

The difference between an easy and difficult ski slope isn't just height; a reasonably competent skier can handle a long, gentle slope. It's those steep ones that tend to be challenging and…entertaining. What makes a hill steep is its rate of change, the change in height for every change in length.

Students should know that to determine the rate of change, they should choose two points and subtract the differences in the height. Divide that number by the differences in horizontal location for those same two points. If this sounds eerily familiar to rise over run to your students, they're eerily perceptive. Slope is the rate of change.

Given a linear function or two points, students should be able to find the rate of change and the initial value (the y-intercept) and interpret it in terms of its context. Remembering that for rate of change, y values go on the numerator and x values go on the denominator (or the far easier "rise over run"). This is essential so that students don't calculate reciprocal slopes.

It's beneficial for students to apply slopes to functions, particularly linear functions. Knowing that the rate of change is constant for a linear function and changes for a nonlinear function is also important. They should know that a positive slope means uphill (the function is increasing) and a negative slope means downhill (the function is decreasing).

Drills

  1. What is the rate of change of a linear function that includes the points (4, 1) and (8, 3)?

    Correct Answer:

    ½

    Answer Explanation:

    Rate of change is how much the height (y value) changes compared to the length (x value). If we set up a ratio that divides the difference of the y coordinates by the difference of the x coordinates, that will give us the rate of change. The y coordinates have a difference of 3 – 1 = 2 and the x coordinates differ by 8 – 4 = 4, so the ratio is 2 ÷ 4 = ½. Other answers result from switching coordinates (subtracting 4 – 8 instead of 8 – 4) or finding the reciprocal of the slope (dividing 4 by 2 instead of 2 by 4).


  2. What is the rate of change of a linear function that includes the points (6, 7) and (10, 11)?

    Correct Answer:

    1

    Answer Explanation:

    If we find the difference between the x and y values and then divide height by length, we should get (11 – 7) ÷ (10 – 6) = 4 ÷ 4 = 1 for the rate of change. Accidentally switching one set of coordinates might result in -1 as the answer and forgetting to use both x and y values might give an incorrect answer of 4 or ¼. The actual slope of the line, however, is 1.


  3. What is the rate of change of a linear function that has the equation 2x + y = 6?

    Correct Answer:

    -2

    Answer Explanation:

    We could either find two points on the line, say (0, 6) and (1, 4) and use them to find the rate of change, or understand that slope and rate of change are the same thing. We can simply rearrange the formula into standard form, y = mx + b, and realize that m is the rate of change. All it takes is subtracting 2x from both sides to get y = -2x + 6. Since m = -2 in the equation, the rate of change is -2 for the function.


  4. What is the rate of change of a linear function that has the equation 2y – 4 = 10x?

    Correct Answer:

    5

    Answer Explanation:

    The rate of change of a linear function is the same as its slope m when the equation is in the form y = mx + b. If we change our equation into that form, we should have y = 5x + 2 and that gives a slope of 5. Other answers may come from not simplifying the equation or forgetting to rewrite it in standard form.


  5. A linear function has a y-intercept of 4 and a point at (-3, 3). What is the rate of change?

    Correct Answer:

    &frac13

    Answer Explanation:

    While this may seem impossibly difficult at first, it helps to know that the y-intercept is really code for the point (0, 4). Now we can subtract the x and y values to calculate the rate of change. If we do it correctly, we get (3 – 4) ÷ ( -3 – 0) = ⅓. Other answer choices are due to misinterpreting the y-intercept or calculating the slope incorrectly.


  6. What does it mean if the rate of change of a linear function is zero?

    Correct Answer:

    The line is horizontal

    Answer Explanation:

    Since a linear function takes the form y = mx + b where m is the slope, a rate of change of zero would mean m = 0. That would make the equation y = 0x + b or just y = b. In other words, the output value will be b regardless of the input value of x. Interpreting this as rate of change (rise over run), the height never increases, which means (A) is right. Horizontal lines are still functions because each x still has one y, whereas vertical lines aren't.


  7. Which of the following is the equation of the linear function containing (0, 1) and (2, 9)?

    Correct Answer:

    y = 4x + 1

    Answer Explanation:

    Of course, we could cheat and just plug in the ordered pairs into all the equations. But since it contains the point (0, 1), the initial value should be 1, which narrows it down to either (A) or (D). And when we find the slope, it turns out to be 4. So we want y = 4x + 1.


  8. Which of the following is the equation of the linear function containing (2, 2) and (6, 6)?

    Correct Answer:

    y = x

    Answer Explanation:

    Start by finding the slope, or rate of change, by finding the ratio of differences between the y and x coordinates. That gives us (6 – 2) ÷ (6 – 2) = 1. The m in y = mx + b has to equal 1, but what about the b? Plugging in the x and y coordinates of either point into the equation should give us b = 0. Our answer should be (B). Any other answer would result from incorrect calculations of slope or y-intercept.


  9. In order to be in the Guinness Book of World Records, Heidi practices blowing her bubblegum every day. She charted her progress of how many bubbles she blew per day for the past year and the result is given by the graph below. Which is true about the initial value of the graph?

    • In order to be in the Guinness Book of World Records, Heidi practices blowing her bubblegum every day. She charted her progress of how many bubbles she blew per day for the past year and the result is given by the graph below. Which is true about the initial value of the graph?

      Correct Answer:

      The initial value is 5, which means Heidi started the year blowing only 5 bubbles

      Answer Explanation:

      The initial value is the same as the y-intercept. They both mean the height at the start of the graph when x = 0. Since the x-axis indicates time and the y-axis shows how many bubbles Heidi blew, the initial value is 5, which means that Heidi blew 5 bubbles before she started her bubblegum training. The answers aside from (C) either give the incorrect number or incorrect definition of initial value in terms of the context.


  10. In order to be in the Guinness Book of World Records, Heidi practices blowing her bubblegum every day. She charted her progress of how many bubbles she blew per day for the past year and the result is given by the graph below. Which is true about the rate of change of the graph?

    Correct Answer:

    The rate of change is 1, which means Heidi blew 1 more bubble with every passing day

    Answer Explanation:

    The rate of change describes how y increases with every x. Since y is the number of bubbles blown and x indicates the day, then we can interpret the rate of change as the number of additional bubbles blown with every passing day. We can calculate this value by choosing another point on the line—any point from (1, 6) to (365, 370) will work—and divide the differences in y by the differences in x. Regardless of the point you choose, you should get a slope of 1. This means (A) is our answer.


More standards from Grade 8 - Functions