Unit 6, Lesson 12
Reasoning about Exponential Graphs (Part 1)
Algebra 1
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Graph of 4 functions on grid, origin O. Horizontal axis, x, from 1 to 8, by 1's. Vertical axis, y, from 0 to 200, by 50's. Functions A, B, C and D all cross the y axis at 0 comma 200, and all move downward and to the right.
Function A passes through 1 comma 175, Function B passes through 1 comma 150, Function C passes through 1 comma 100, Function D passes through 1 comma 50.
Match each equation with a graph. Be prepared to explain your reasoning.
Functions and are defined by these two equations: and .
An exponential function can give us information about a graph that represents it.
For example, suppose that function represents a bacteria population hours after it is first measured, and . The number 5,000 is the bacteria population measured, when is 0. The number 1.5 indicates that the bacteria population increases by a factor of 1.5 each hour.
A graph can help us see how the starting population (5,000) and growth factor (1.5) influence the population. Suppose functions and represent two other bacteria populations and are given by and . Here are the graphs of , , and .
Graph of 3 functions labeled, p, q, and r on grid, origin O. Horizontal axis, time in hours, from 0 to 10, by 2's. Vertical axis, number of bacteria, from 0 to 40,000 by 20,000's. All three functions start at 0 comma 5,000 and trend upward and to the right. Function p passes through 2 comma 20,000 and trends rapidly upward and right. Function q passes through 4 comma11,250 and trends steadily upward and right. Function r passes through 6 comma 7,200 and trends more slowly upward and right.
All three graphs start at , but the graph of grows more slowly than does the graph of , while the graph of grows more quickly. This makes sense because a population that doubles every hour is growing more quickly than one that increases by a factor of 1.5 each hour, and both grow more quickly than a population that increases by a factor of 1.2 each hour.