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Common Core Standards: Math

High School: Algebra

Reasoning with Equations and Inequalities HSA-REI.D.11

11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Students should understand that an equation and its graph are just two different representations of the same thing. The graph of the line or curve of a two-variable equation shows in visual form all of the solutions (infinite as they may be) to our equation in written form. When two equations are set to equal one another, their solution is the point at which graphically they intersect one another. Depending on the equations (and the alignment of the planets), there might be one solution, or more, or none at all.

Students can arrive at the correct answer(s) through graphing the functions and plotting their intersection points, creating a table of x and f(x) values, and solving for x algebraically when f(x) = g(x). These strategies should be provided to students and practiced with students so that the connection between graphs and equations are solidified. (We wouldn't want them to be liquefied, now would we?)

Drills

  1. At which point do the two equations 3x + 5 = y + 4x and y = x2 intersect?

    Correct Answer:

    Both (A) and (B)

    Answer Explanation:

    We can solve the answer by simplifying our first equation to y = 5 – x and then setting the two equations equal to one another. That means x2 = 5 – x and simplifies to x2 + x – 5 = 0 and we can use the quadratic formula to solve for x. The two values for x are 1.8 and -2.8, with the y values at 3.2 and 7.8, respectively. If we graph the two functions, we can see that these answers make sense.


  2. At which point do the equations y = |x + 3| and 15x + 20y = 0 intersect?

    Correct Answer:

    Both (A) and (B)

    Answer Explanation:

    Graphing the two functions, we can see that they intersect within the negative x region in two points. It is fairly obvious that one of these points has the coordinates of (B), but there is another point closer to the origin with the points of (A). While the point (0, 3) is valid for the equation |x + 3| = y, it does not satisfy the equation 15x + 20y = 0.


  3. Find the intersection point of y = log x and y = 3x.

    Correct Answer:

    There is no intersection

    Answer Explanation:

    If we graph the two equations, it will be clear that the two functions never intersect. Plugging in every point into both equations and seeing that none work will give the same logical conclusion.


  4. Find the intersection of the two equations y = 17x + 1 and y = -24x + 68.

    Correct Answer:

    (1.6, 29)

    Answer Explanation:

    After graphing the two equations we can approximate the point of intersect at about (1.6, 29). We can also set the two equations equal to each other to find the exact values.


  5. Imagine you're interviewing for a position with Shmoop (your dream job, obviously) and you don't want to screw it up. Dave, the interviewer, asks you to find the intersection of the equations y = -x + 8 and y = -2x + 16. What do you tell him?

    Correct Answer:

    (8, 0)

    Answer Explanation:

    First, you ask Dave for a pencil and some graph paper. Then, you graph the two lines and you see that their intersection point is at (8, 0). You show it to Dave, shake his hand, and wait for your offer letter.


  6. Which of the following points is on both line y = -x + 2 and line y = x + 2?

    Correct Answer:

    (0, 2)

    Answer Explanation:

    For such simple equations, we suggest you start with a table of values with integers for x ranging from -2 to 2 and see what you get for each equation. We can see from the table below that the y values are equal when x is equal to 0. Hence (0, 2) is our point of intersection.

    xy = -x + 2y = x + 2
    -240
    -131
    022
    113
    204

  7. When estimating the intersection of two lines on a graph, you can get a precise answer. Is this statement true or false?

    Correct Answer:

    Both (B) and (C)

    Answer Explanation:

    Using a graph to find the point of intersection is highly reliant on what area of the graph you're looking at, how much you zoomed in, how sensitive the scale is, and so on. Hence, it is always an approximation until you confirm the intersection using mathematical means!


  8. Your classmate needs to find the points of intersection of two very simple equations. She makes a table that lists 5 or so integers and finds a point of intersection. She thinks she's done and goes to play outside. What do you think of this?

    Correct Answer:

    A table of values may not show the full story because there may be points of intersection missed, but they're easy equations so she's probably done.

    Answer Explanation:

    While a couple of easy equations (like two linear equations) may only have one point of intersection, it's best not to develop a habit of thinking there is always one. To be safe, either graph the equation or mathematically solve it help you find the number of solutions to a system.


  9. When you find a point on a graph that you think is the intersection, the best way to double check your answer is to:

    Correct Answer:

    Plug in the x value to see if both of the y values of both of the equations match the one you approximated from the graph

    Answer Explanation:

    Both y values must match the y value you derived from you graph in order to ensure an accurate point of intersection. This is because both of the equations must come up with the same x and y values (otherwise they aren't the same point).


  10. The most accurate way of finding the points of intersection of a system of equations is via:

    Correct Answer:

    Using algebra to solve for the variables

    Answer Explanation:

    Answers (A) and (B), while useful in terms of visualization, are not as thorough as algebraically solving for the variables. They can lead to inaccurate or completely missed points of intersection, while algebra (if done correctly, of course), will never lead you astray.