Math 1 · Unit 7 · Lesson 4 · part 2 of 6
Four key features, and where each comes from
In Unit 3, Lesson 3 you learned to read a line's graph: where it crosses, which way it tilts.
Four key features, and where each comes from
In Unit 3, Lesson 3 you learned to read a line's graph: where it crosses, which way it tilts. An exponential graph has its own list of key features, and every one of them comes from a or b.
- 1The y-intercept is (0, a).Lesson 2: b0 = 1, so at x = 0 the output is a. The curve always crosses the y-axis at a.
- 2Domain: all real numbers.You can put in any x, positive, negative, or a fraction. The table walks both ways forever (Lesson 1). The curve never ends left or right.
- 3Range: y is greater than 0 (when a is positive).The output is a positive number multiplied by positive factors, so it stays positive. It can get very small but it is never 0 and never negative.
- 4The horizontal asymptote is the line y = 0.An asymptote is a line the graph gets closer and closer to without ever touching. Walk the doubling table left: 1, one half, one quarter, one eighth. Each step halves the height, so it keeps shrinking toward the x-axis but never reaches it. The x-axis, y = 0, is the asymptote.
Why the graph never touches the x-axisTo reach 0 you would have to multiply something by 0. But b is positive, and a positive number times a positive number is positive. Walking left divides by b every step, and dividing a positive number never gives 0: 11000, 11000000, smaller and smaller, but always more than 0.