Two-variable data: models and scatterplots

Most of these questions show a scatterplot, with one dot for each pair of measurements and often a line of best fit drawn through them. Others give a curve, a table, or a short story with no graph.

Read the last line first. It might say:

• “Which of the following is closest to the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson of the line of best fit?”
• “Which of the following is the best interpretation of the slope of the line of best fit in this context?”
• “How many of the data points lie above the line of best fit?”
• “Which of the following equationsTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson is the most appropriate exponentialA relationship where each step multiplies by the same number, like doubling every year.Read the lesson model for the data shown in the scatterplot?”
• “What type of functionA rule that takes in one number and gives back one number, like f(x) = 5x - 7.Read the lesson best models the relationship?”

There's no way to type a picture's dots into Desmos, so it helps only with a table, or to check a model against two points.

Before you read a number off a graph, read both axisOne of the two lines a graph is measured against, one running across and one running up. labels so you know what each one measures and in what unit. Then check what a square is worth.

An axis might also start above 0, so the bottom edge of the graph isn't 0. And the label might say “in hundreds of liters.” Then a reading of 3 means 300 liters, and a question about a 600-liter tank sends you to 6 on that axis.

At a charity walk, each walker's dot shows how many laps they walked and how much money they raised. Each square across is 1 lap, and each square up is $4.

Find 3 laps on the horizontal axis. The dot above it is at $24, and the line passes through $20 there.

Some questions ask how many dots sit above or below the line. Go left to right and compare each dot with the line directly above or below it. On the walk's plot, 4 dots are above the line and 4 are below. The dots at 4 and 8 laps sit only half a square above it, so look closely there. A dot right on the line counts as neither.

Here's one where the scale matters.

+−100%Reset⤢

The scatterplot shows completed pages during Layla’s print run and a line of best fit.

How many pages had been completed after $7$ minutes?

1. A

   $37.5$

2. B

   $40$

3. C

   $7$

4. D

   $9.5$

Show the answer

• A

  $37.5$

  This reads the line’s predicted value of $7.5$ groups at $7$ minutes instead of the observed dot. Converting that prediction gives $7.5 \times 5 = 37.5$ pages.

• B

  $40$

  Correct answer

  At $7$ minutes, the observed dot is at $8$ groups of $5$ pages. Therefore, $8 \times 5 = 40$ pages had been completed.

• C

  $7$

  This reports the observed dot’s horizontal coordinate, $7$, instead of reading its vertical coordinate of $8$ groups.

• D

  $9.5$

  This reads $9.5$ groups from the neighboring point at $8$ minutes. It also treats each vertical-axis unit as $1$ page even though each unit represents $5$ pages.

Solution

Find the observed dot whose horizontal coordinate is $7$ minutes. Its vertical coordinate is $8$, so Layla completed $8$ groups of pages.

Each group contains $5$ pages, so $8 \times 5 = 40$. After $7$ minutes, $40$ pages had been completed.

Report a problem

On the walk's plot, the 3-lap walker raised $24 and the line predicted $20. “How much more than predicted” means the actual value minus the predicted one. This difference is called the residualThe actual value minus the value the line of best fit predicts, so a point above the line gives a positive number..

A question might ask where the line predicts more than was recorded. That's every dot below the line.

You find the slope the same way as in linear functions. On a scatterplot, the work is in picking two points the line itself crosses at gridlines. On the walk's plot, the line meets the vertical axis at $8 and passes through the corner where 5 laps meets $28.

1. Two places the line crosses gridlines.

   $$(0,8) \text{ and } (5,28)$$

2. Change in dollars over change in laps.

   $$m=\frac{28-8}{5-0}$$

3. Divide. The line rises $4 a lap.

   $$m=\frac{20}{5}=4$$

The line won't always reach the vertical axis. Say it crosses gridlines only at $(2,9)$ and $(10,29)$. Those two points give the slope, and walking back to $x=0$ gives the y-interceptThe output when the input is 0, which is where a graph crosses the vertical axis.Read the lesson.

1. Change in $y$ over change in $x$.

   $$m=\frac{29-9}{10-2}$$

2. Divide.

   $$m=\frac{20}{8}=2.5$$

3. From $(2,9)$, go 2 units left to $x=0$. The line drops 2.5 for each unit left.

   $$9-2.5(2)=4$$

4. Write the equation.

   $$y=2.5x+4$$

The walk's slope is 4, in dollars per lap. To say what it means, put both quantities in the sentence.

A line of best fit for pizza prices at several shops is $y=2x+13$, where $x$ is the number of toppings and $y$ is the price in dollars.

If none of the pizzas in the data were plain, 13 is only where the line lands when you follow it back to $x=0$. It still means the predicted price of a plain pizza, so pick the choice that calls it predicted or estimated.

Here's one with a falling line.

+−100%Reset⤢

Farid tracks unsold bus-tour seats after ticket sales begin.

Which of the following is the best interpretation of the slope of the line of best fit in this context?

1. A

   The predicted number of days after ticket sales begin decreases by $0.5$ for each additional unsold seat.

2. B

   The predicted number of unsold seats decreases by $0.5$ for each additional day after ticket sales begin.

3. C

   The predicted number of unsold seats increases by $0.5$ for each additional day after ticket sales begin.

4. D

   The predicted number of unsold seats decreases by $9$ for each additional day after ticket sales begin.

Show the answer

• A

  The predicted number of days after ticket sales begin decreases by $0.5$ for each additional unsold seat.

  This interpretation exchanges the inputThe number you put into a function, like the 4 in f(4).Read the lesson and outputThe number a function gives back, like the 13 when f(4) = 13.Read the lesson quantities. The slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson describes predicted unsold seats per additional day, not predicted days per additional unsold seat.

• B

  The predicted number of unsold seats decreases by $0.5$ for each additional day after ticket sales begin.

  Correct answer

  Using two points on the line of best fitThe straight line that best follows the trend of the dots in a scatterplot.Read the lesson, the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson is $\frac{2-9}{14-0}=-0.5$. Therefore, the predicted number of unsold seats decreases by $0.5$ for each additional day after ticket sales begin.

• C

  The predicted number of unsold seats increases by $0.5$ for each additional day after ticket sales begin.

  This interpretation flips the sign of the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson. Because the line falls from left to right, its slope is negative, so the predicted number of unsold seats decreases.

• D

  The predicted number of unsold seats decreases by $9$ for each additional day after ticket sales begin.

  This uses the $y$-intercept, $9$, as the per-day rate of changeHow much the output changes each time the input goes up by 1.Read the lesson. The interceptWhere a graph crosses an axis.Read the lesson is the predicted number of unsold seats when ticket sales begin, while the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson is $-0.5$ seat per day.

Solution

Choose two points on the line of best fitThe straight line that best follows the trend of the dots in a scatterplot.Read the lesson, such as $(0,9)$ and $(14,2)$. Compute the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson:

$\frac{2-9}{14-0}=\frac{-7}{14}=-0.5$.

The horizontal quantity is days after ticket sales began, and the vertical quantity is unsold seats. A slope of $-0.5$ means that for each additional day, the predicted number of unsold seats decreases by $0.5$.

Report a problem

An average rate of changeHow much the output changes each time the input goes up by 1.Read the lesson comes from the two dots the question names. Say a question asks for it from $x=2$ to $x=6$ on the walk's plot, where the 2-lap dot is at $14 and the 6-lap dot is at $26.

1. Read the two named dots.

   $$(2,14) \text{ and } (6,26)$$

2. Change in dollars over change in laps.

   $$\frac{26-14}{6-2}$$

3. Divide. That's $3 per lap.

   $$=\frac{12}{4}=3$$

The dots in between, 24, 26 and 32, don't change the answer, and neither does the line's slope of 4.

When the later dot is lower, the rate is negative. From $32 at 5 laps to $26 at 6 laps, the rate is $\frac{26-32}{6-5}=-6$ dollars per lap. On a grid-inA question where you type your own answer instead of picking one of four.Read the lesson, type the minus sign: -6.

Try one where the dots fall.

+−100%Reset⤢

Fabre recorded the number of flower stems used in one display on each of seven days. The scatterplot shows the data.

What was the average rate of change, in stems per day, from day 2 to day 8?

Show the answer

Accepted answers: -1/2, -0.5, -.5

Solution

From day 2 to day 8, the number of stems changes from 25 to 22, so the change is $22-25=-3$ stems over $8-2=6$ days. The average rate of changeHow much the output changes each time the input goes up by 1.Read the lesson is $\frac{-3}{6}=-\frac{1}{2}$ stem per day.

A response of $-3$ gives the total change in stems without dividing by the 6 elapsed days. A response of $0.5$ uses the correct magnitude but drops the minus sign that shows the number of stems decreased.

Note that $-1/2$, $-0.5$, and $-.5$ are examples of ways to enter a correct answer.

Report a problem

To describe a graph or a list of values, answer two questions separately. Do the values go up or down as $x$ goes up? And does each step add the same amount, or multiply by the same number?

For the shape, use equally spaced $x$-values and compare each $y$-value with the next.

1. Values at $x=0,1,2,3$.

   $$200,\ 300,\ 450,\ 675$$

2. Subtract each value from the next. The differences change, so it isn't linear.

   $$100,\ 150,\ 225$$

3. Divide each value by the one before. It's 1.5 every time.

   $$\frac{300}{200}=\frac{450}{300}=\frac{675}{450}=1.5$$

A quadraticAn expression whose biggest power is a square, like x squared plus 3x minus 4.Read the lesson model turns around. Dots that rise, level off and come back down can't fit a linear or an exponential model, because neither one ever turns. In a table, check the differences of the differences. For 1, 4, 9, 16 the differences are 3, 5, 7, and their differences are 2 and 2. When those match, the model is quadratic.

A story with no graph works the same way. A snowbank that loses 2 inches of depth each day is decreasing linear, because it loses the same amount every day. A used phone that loses 15% of its value each year is decreasing exponential, because it loses the same percent. Doubling and halving are exponential too.

An exponential model is written $y=a(b)^x$. $a$ is the starting value, the $y$-value at $x=0$, and $b$ is the multiplier for each step of 1 in $x$. Say a falling curve passes through $(0,90)$ and $(1,45)$. Then $a=90$, and $b=\frac{45}{90}=0.5$.

A $b$ above 1 makes the curve grow, and a $b$ between 0 and 1 makes it shrink. So $y=90(2)^x$ can't fit a falling curve.

The two marked points might be more than one step apart, like $(0,100)$ and $(2,81)$.

1. Divide the two $y$-values. This is the multiplier for 2 steps.

   $$\frac{81}{100}=0.81$$

2. Two steps means multiplying by $b$ twice.

   $$b\cdot b=0.81$$

3. Take the square root of the top and of the bottom.

   $$b=\sqrt{\frac{81}{100}}=\frac{9}{10}$$

4. Divide.

   $$b=0.9$$

5. Check: two steps from 100 land on 81.

   $$100(0.9)(0.9)=81$$

A table works the same way. If $x$ goes 0, 2, 4, 6 and the values go 5, 10, 20, 40, each row doubles over 2 steps of $x$. So $b=\sqrt{2}$, about 1.41, and you can also write the model as $y=5(2)^{x/2}$.

When the data come as a table, Desmos can find the slope and starting value of a linear model for you.

For an exponential or quadratic choice, use the check in the last section at two of the table's $x$-values. Desmos for data shows more of this.

Say a data set has the line of best fit $y=4x+6$, and a new data set is made by multiplying every $y$-value by 2.5. Every prediction gets 2.5 times bigger, including the one at $x=0$.

1. Multiply the whole right side by 2.5.

   $$y=2.5(4x+6)$$

2. $2.5\cdot4=10$ and $2.5\cdot6=15$.

   $$y=10x+15$$

If every $x$-value is doubled instead, each rise takes twice as far across, so the slope is cut in half. $y=4x+6$ becomes $y=2x+6$. The starting value stays 6, since doubling $x=0$ still gives 0.

If the first line is drawn on a graph, read its slope and starting value, then multiply.

Before each one, decide whether the question wants a dot, the line, or the pattern in the values. Then check that the sign of your answer matches the direction of the graph.