On the Digital SAT, slope questions are considered the heart of basic Algebra. If you had to pick just one thing from Algebra 1 to remember, the SAT would prefer that it be everything about slope. It’s a rate of change of an equation, the change in y over the change in x, rise over run, and the m in y = mx + b. Under the pressure of the SAT, it can be easy to momentarily flip-flop key details of slope – did I solve it upside-down or backwards? Should it be negative or positive here? Any problem that raises this kind of doubt is a great candidate for support from the Desmos calculator. You can either confirm your work with it, or just let it do a lot of the solving for you!
You may also want to check out our general Desmos strategies and the rest of the Desmos series.
Same Slope Problem
Some questions will ask you to pick an answer choice that includes an equation with the same slope as one in the prompt. If a linear equation is written in slope-intercept form – y = mx + b – then the slope is the value of m and the question is simple. The SAT will therefore make things tricky by writing equations in other equivalent forms that disguise the slope. You can skip the algebra and the uncertainty by comparing lines in Desmos: the line with the same slope is going to be the one that is parallel to the original.
Desmos Trick:
- Type the original equation into desmos. You can type exactly what the problem says, even if it is not solved for y.
- Type each of the possible solutions from the answer choices into Desmos.
- Each of the equations you write should appear on the graph as a straight line.
- Compare each choice to the original equation.
Example One (match the slope):
Which of the following equations has the same slope as the given equation?
There are always a few interesting ideas in the wrong answers of SAT questions, and that is what makes each one of these a little tempting in its own way– until we really do the math. Any chance of misremembering or going too fast can be avoided by graphing each option in Desmos.
Step 1: The original equation’s graph:
Step 2: Then you graph all of the answer choices:
Step 3: You can see that the blue line, representing answer choice A, is parallel to the red line, which was the original equation. That means choice A had the same slope and is our answer!
Example Two (Slope Transformation with Function):
Given the following function:
Which of the following equations has the same slope as 2g(x)?
Step 1: Graph g(x) and then use the next line to graph 2g(x) – yes, Desmos can do that! The blue line will be our target for the rest of the problem.
Step 2: Graph all of the other equations. You may notice that Desmos has only a few colors and so they may repeat, and you will have to be careful about distinguishing those lines from one another.
Step 3: It looks like we have two parallel blue lines, and one of them is 2g(x) (the target of our question), while the other is coming from answer choice C. The answer to this question is C.
Example 3 (Slope and point comparison):
Line p in the xy-plane has a slope of ½ and passes through the point (4, -3). Which equation defines line p?
Step 1: Graph the point. Add a label if you like.
Step 2: Graph all of the lines.
Step 3: This time we have to do a little interpreting. Two lines go through the point we labeled. Only one has a slope of ½. We need to come to this problem with the knowledge that a positive slope goes up when read from left to right, which means that the blue line representing answer choice D is correct.
Example 4 (perpendicular lines question):
The line defined by y = f(x) = ⅔ x + 12 is perpendicular to line m, and the two lines intersect at the point (r, 0), where r is a constant. What is the equation of line m?
Step 1: This is a lot like the “same slope” questions, but now we have the key term perpendicular. We’ll graph the original line.
Perpendicular: two lines that meet at a 90° (right) angle. In equation terms, perpendicular lines have slopes with that are opposite-signed reciprocals. For example, a line with a slope of ¾ is perpendicular to one with a slope of -4/3.
Step 2: Graph all of the answer choices. Remember that the original equation is represented by the red one, and this time we want to find a particular perpendicular line.
Step 3: If you already knew the concept of perpendicular very well, you might have spotted that only answer choice C has the opposite reciprocal slope without even graphing. This graph helps to visually confirm it! We can see that the green line meets the red one at a nice clear 90 degree angle, and it also intersects at the x-intercept of both respective lines– that’s what the somewhat confusing language of meeting at point (r, 0) was about. So for all these reasons, the answer is definitely C.
Practice Problems
- If the system of the following two equations has infinite solutions, what must be the value of k?
2. The line y=ax+b is perpendicular to the line y=cx+d , where a, b, c and d are integer constants. Which of the following must be true?
3. Line p has the equation 2x + 3y = 6. If line p intersects with line m at exactly one point, which of the following could be the equation of line m?
Answers:
1) 9/4 or 2.25
2) C
3) C
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