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Study Guides > Finite Math

Reading: Graphs of Linear Functions (part II)

Intersections of Lines

The graphs of two lines will intersect if they are not parallel. They will intersect at the point that satisfies both equations. To find this point when the equations are given as functions, we can solve for an input value so that f(x) = g(x). In other words, we can set the formulas for the lines equal, and solve for the input that satisfies the equation.

Example 13

Find the intersection of the lines and h(t) = 3t – 4 and j(t) = 5 – t Setting h(t) = j(t), 3t - 4 = 5-t 4t = 9 $latex t = \frac{9}{4} $ This tells us the lines intersect when the input is $latex t = \frac{9}{4} $ We can then find the output value of the intersection point by evaluating either function at this input $latex j(\frac{9}{4})=5-\frac{9}{4}=\frac{11}{4}$ These lines intersect at the point $latex (\frac{9}{4},\frac{11}{4}) $ . Looking at the graph, this result seems reasonable. Two parallel lines can also intersect if they happen to be the same line. In that case, they intersect at every point on the lines.

Try it Now 4

Look at the graph in example 13 above and answer the following for the function j(t):
  1. Vertical intercept coordinates
  2. Horizontal intercepts coordinates
  3. Slope
  4. Is j(t) parallel or perpendicular to h(t) (or neither)
  5. Is j(t) an Increasing or Decreasing function (or neither)
  6. Write a transformation description from the identity toolkit function f(x) = x
Finding the intersection allows us to answer other questions as well, such as discovering when one function is larger than another.

Example 14

Using the functions from the previous example, for what values of t is h(t) > j(t) To answer this question, it is helpful first to know where the functions are equal, since that is the point where h(t) could switch from being greater to smaller than j(t) or vice-versa. From the previous example, we know the functions are equal at$latex t = \frac{9}{4} $. By examining the graph, we can see that h(t), the function with positive slope, is going to be larger than the other function to the right of the intersection. So when

Important Topics of this Section

  • Intersecting lines

Try it Now 4 Answers

  1. (0,5)
  2. (5,0)
  3. Slope –1
  4. Neither parallel nor perpendicular
  5. Decreasing function
  6. Given the identity function, perform a vertical flip (over the t axis) and shift up 5 units.
David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, "Chapter 2: Linear Functions," licensed under a CC BY-SA 3.0 license.