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Two-variable data: Models and scatterplots: SAT Math notes
These questions test whether you can read scatterplots and line graphs, use a line or curve of best fit to make predictions, interpret its slope, and choose the kind of model (linear, quadratic or exponential) that fits the data.
Study it with worked examples and 7 practice questionsPractise Two-variable data: Models and scatterplots questions
What this topic tests
Two-variable data pair up two measurements, like the temperature and the number of beach visitors on the same day. A scatterplot puts one on each axis, and a line or curve of best fit shows the overall trend, even though the individual points don't sit exactly on it.
Most of these questions come with a graph, so reading it accurately is half the battle. Check the axis labels, the units and what each grid line is worth before you read anything off it. Then connect the picture to the algebra: the slope of the line is a rate, and the y-intercept is a starting value.
A few questions have no graph at all. They describe a situation and ask whether it changes by the same amount each step (linear) or by the same percent each step (exponential).
Key ideas
- A line of best fit gives predicted values. To predict, find the x-value on the horizontal axis, go up to the line, then across to the vertical axis.
- The gap between an actual data point and the line's prediction at the same x is called a residual: actual minus predicted.
- The slope of a best-fit line is the predicted change in y for each increase of 1 in x, measured in the units of the axes.
- Points rising from left to right show a positive association and a positive slope. Points falling show a negative association and a negative slope.
- To match an equation to a graph, check the sign of the slope, then the y-intercept (the value at x=0), then how steep the slope is.
- Linear models add the same amount each step. Exponential models multiply by the same factor each step: y=a(b)x grows when b>1 and decays when 0<b<1.
- A curve that rises then falls (an upside-down U) is a quadratic with a negative x2 coefficient. A U shape has a positive one.
The 4 kinds of Two-variable data: Models and scatterplots question
Sorted from every Two-variable data: Models and scatterplots question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Read a value or prediction from a graph
How to spot it. The question points you to one spot on a graph: the value the line of best fit predicts at a given x, the gap between a data point and the line, the lowest point on a line graph, or the average of a few points in one region.
How to answer it:
- Read the axis labels and work out what one grid square is worth on each axis.
- Find the given x-value on the horizontal axis and move straight up to the line, curve or point you need.
- Move straight across to the vertical axis and read the value, estimating between grid lines if needed.
- For a residual, read the actual point and the line at the same x, then subtract.
The trap. Reading a data point when the question wants the line's prediction, or the other way round. According to the line of best fit means use the line, even if a dot is close by. Also check the scale: grid lines might count by 2s, 50s or 80s.
2. Slope and rate of change
How to spot it. The question asks for the slope of a best-fit line, what the slope means in context, the predicted change in y for some change in x, or where a graph rises or falls fastest.
How to answer it:
- Pick two points that are clearly on the line (not data points), as far apart as you can and on grid lines if possible.
- Slope = change in y ÷ change in x. If the line falls, the slope is negative.
- Attach units: the slope is y-units per one x-unit. Say it as a sentence: for each extra 1 of x, y is predicted to change by the slope.
- For a change of k units in x, multiply the slope by k.
The trap. Mixing up the variables in an interpretation. If x is seeds planted and y is seeds that germinated, the slope is germinated seeds per seed planted; it says nothing about time. Also remember the slope describes the trend, not an exact rule for every data point.
3. Choose the equation that fits the data
How to spot it. You see a scatterplot and four equations, and you're asked which one could be the line or curve of best fit. The choices often differ only in their signs, in which number is the slope, or in the kind of model.
How to answer it:
- Decide the shape: a straight trend is linear, a U or upside-down U is quadratic, and a curve that keeps getting steeper (or flatter) is exponential.
- Check the direction: rising means a positive slope or growth, falling means a negative slope or decay. Cross off choices with the wrong sign.
- Check the starting value: where the trend meets x=0 should match the constant term.
- If two choices are still left, estimate the slope from two points on the line, or plug an x-value from the graph into each choice.
The trap. Swapping the slope and the intercept. y=0.82x+3.30 and y=3.30x+0.82 use the same numbers but describe very different lines. The number multiplying x must match how steep the data are, and the constant must match the starting value.
4. Linear or exponential?
How to spot it. Often there's no graph. A situation or table is described and you must decide whether a quantity changes by a constant amount (linear) or a constant percent (exponential), or compare a linear quantity with an exponential one over time.
How to answer it:
- Ask how the quantity changes each step. The same amount added or subtracted each time is linear. The same percent or factor each time is exponential.
- Decide the direction: going up means a positive slope or growth, going down means a negative slope or decay.
- In a table, look at the differences between consecutive y-values. Equal differences mean linear; equal ratios mean exponential.
- When comparing a linear and an exponential quantity, test a few values or graph both. A growing exponential always overtakes a line eventually.
The trap. Calling anything that goes down exponential decay. Checking shirts at a constant rate removes the same number each hour, so the number left falls in a straight line. Exponential decay needs the same percent lost each step.
Practise Two-variable data: Models and scatterplots
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