Graphs of Functions in Standard and Factored Forms

5 min

Narrative

In this Warm-up, students revisit the graph of a linear equation and recall that an equation can tell us something about the graph that represents it and vice versa.

Student Task

Here is a graph of the equation y=8−2xy = 8-2x.

<p><strong>Graph of line, origin O. Horizontal x axis from negative 2 to 5 by 1’s. Vertical y axis from negative 2 to 12, by 2’s. Line starts at 0 comma 8, passes through 1 comma 6, 2 comma 4, and 4 comma 0.</strong></p>

  1. Where do you see the 8 from the equation in the graph?
  2. Where do you see the -2 from the equation in the graph?
  3. What is the xx-intercept of the graph? How does this relate to the equation?

Sample Response

  1. The 8 in y=8−2xy = 8 - 2x means that the graph intersects the yy-axis at 8. The yy-intercept is (0,8)(0,8).
  2. The -2 is the slope of the line. You can see it, for example, by looking at (0,8)(0,8) and (1,6)(1,6). When xx increases by 1, yy decreases by 2.
  3. The xx-intercept is (4,0)(4,0). The 4 is the solution of 8−2x=08 - 2x = 0.

Synthesis

Make sure students recall that:

  • The yy-intercept tells us where a graph intersects the yy-axis, at which point the xx value is 0. When a linear equation is written in the form y=mx+by=mx+b, the bb tells us the yy-intercept because when xx is 0, mxmx is also 0, so y=by=b and (0,b)(0,b) is the yy-intercept.
  • The xx-intercept tells us where a graph intersects the xx-axis, at which point the yy value is 0. To find where the graph intersects the xx-axis, find the solution to 0=mx+b0 = mx + b.

Tell students that, when written in certain forms, quadratic expressions also tell us information about their graphs, and vice versa. We’ll explore these connections in this lesson and upcoming ones.

Standards
Building On
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>

15 min

15 min