Topic 1: Graphs and Equations of Exponential Functions

I can describe the characteristics of an exponential function by analyzing its graph. I can describe the characteristics of an exponential function by analyzing its equation. I can match equations in a given set to their corresponding graphs. Interpret the graph of an exponential function that models a situation, and explain the reasoning.

Explore… Use technology to draw the graph of the polynomial functions below. Set your windows to X: [–10, 10, 1] and Y: [0, 10, 1]. Complete the table in each section. Work through all of the Explore activity before proceeding to the next slides, which will outline what you should have noticed…

You Should Notice… Part A The parameter a is equal to the y-intercept. The parameter b is equal to the constant ratio. The graphs both have 1 y-intercept and 0 x-intercepts They are both increasing. They all have a domain of {x|x∈R} and a range of {y|y>0, y∈R}

You Should Notice… Part B The parameter a is equal to the y-intercept. The parameter b is equal to the constant ratio. The graphs both have 1 y-intercept and 0 x-intercepts They are both decreasing. They both have a domain of {x|x∈R} and a range of {y|y>0, y∈R}

You Should Notice… In General (Part C) The parameter a is equal to the y-intercept. The parameter b is equal to the constant ratio. If b is greater than 1 the graph will be increasing If b is between 0 and 1 the graph will be decreasing The graphs all have 1 y-intercept and 0 x-intercepts They all have a domain of {x|x∈R} and a range of {y|y>0, y∈R}. They all (for Math 30-2) extend from Q2 Q1.

Information An exponential function has the form , where x is the exponent b > 0 b ≠ 1 In Math 30-2, the a-value will always be positive The ratio that exists between y-values that correspond to consecutive x-values is known as the common ratio.

Information The graphs of exponential functions have many characteristics. The characteristics that will be explored in this topic are the number of x-intercepts, the value of the y-intercept, domain, range, and end behaviour. The end behaviour of a function is the description of the graph’s behaviour at the far left and the far right.

Example 1 a, b, and c are exponential functions. They all have x as the exponent, and a b value greater than 0 (but not 1). Identifying an exponential function Identify which of the following functions is an exponential function. Explain. a) b) c) d) e) f) This one is not an exponential function since the base (b-value)is negative. This one is not an exponential function since there is no variable. It is a constant function. This one is not an exponential function since the base (b-value)contains a variable (x).

Example 2 Determine if a table of values represents an exponential function Identify which of the following tables of values represent an exponential function. Explain. a) b) Note: To identify whether or not a set of data values represents an exponential function, you must see a common ratio. Ensure that the x-values are increasing by a constant value. Check if there is a common ratio (multiplier) between y-values. If so, the data represents an exponential function.

Example 2 Determine if a table of values represents an exponential function Identify which of the following tables of values represent an exponential function. For those that do, create an equation that represents the function. a) b) x-values are increasing by 1. Each value is obtained by adding 2 to the previous y-value. Since we are adding instead of multiplying, this does not show an exponential function. x-values are increasing by 1. Each value is obtained by multiplying the previous y-value by 2. Since we are multiplying by a common ratio (2), the data represents an exponential function.

Example 2 Determine if a table of values represents an exponential function Identify which of the following tables of values represent an exponential function. For those that do, create an equation that represents the function. c) d) x-values are increasing by 1. There is no common ratio, so this data does not represent an exponential function. x-values are increasing by 1. Each value is obtained by multiplying the previous y-value by 1/2. Since we are multiplying by a common ratio, the data represents an exponential function.

Example 3 Using an equation to determine characteristics of a graph On the next slide, determine the characteristics of the given exponential functions, when a > 0.

Example 3 a = 1 a = 9 b = 2 3 b = e none none 1 9 Q2  Q1 Q2  Q1 increasing decreasing {x|x ∈ R} {x|x ∈ R} {y|y>0, 𝑦∈ R} {y|y >0, 𝑦∈ R}

Example 3 c) Which characteristics are common to all exponential functions with positive a-values. Domain: {x|x ∈ R} Range: {y|y>0, y ∈ R} End Behaviour – Q2  Q1 a = y-intercept No x-intercepts

Example 4 Answers will vary! Reasoning about the characteristics of the graphs of exponential functions For each set of characteristics below, write an equation of a possible exponential function. a) y-intercept of 3 decreasing exponential function b) y-intercept of 1 increasing exponential function c) y-intercept of 7 end behaviour extends from quadrant II to quadrant I 𝑦=3 1 2 𝑥 Note: The y-intercept is given by the a-value. Note: The b-value tells us that the function is increasing (b >1) or decreasing (0 < b < 1). 𝑦=1 3 𝑥 𝑦=7 1 2 𝑥

Example 5 Matching exponential functions to their graphs Match each graph with the correct exponential function. Justify your reasoning. 𝒚=𝟐 𝟒 𝒙 𝒚=𝟑 𝟎.𝟐 𝒙 This graph has a y-intercept (a-value) or 2 and is increasing (b-value is greater than 1). This graph has y-intercept (a-value) of 3, and is decreasing (b-value is between 0 and 1).

Example 5 Matching exponential functions to their graphs Match each graph with the correct exponential function. Justify your reasoning. 𝒚=𝟒 𝟑 𝒙 𝒚=𝟒 𝟎.𝟓 𝒙 This graph has a y-intercept (a-value) of 4, and is increasing (b-value is greater than 1). This graph has a y-intercept (a-value) of 4, and is decreasing (b-value is between 0 and 1).

Example 6 Recognizing an exponential function How can you recognize when a graph, a table of values, or an equation represents an exponential function? From a Table of Values: There must be a common ratio. Ensure that the x-values are increasing by 1. Check if there is a common ratio (multiplier) between y-values. From a Graph: Graph characteristics There must be a smooth curve extending from Q2  Q1 with a domain {x|x∈R} and range {y|y>0, y∈R}. From a Equation: Equation characteristics The equation must be of the form y = abx , where b is positive (but not 1).

Need to Know An exponential function has the form f(x) = a(b)x, where x is the exponent, b > 0, and b ≠ 1. In Math 30-2, all exponential functions have a > 0. The parameter a is the y-intercept, where a > 0. The parameter b is the base, where b > 0 and b ≠ 1. It is the constant ratio, in a table of values, between consecutive y-values when the x-values increase by the same amount. If b > 1, then the exponential function is increasing. If 0 < b < 1, then the exponential function is decreasing.

Need to Know All exponential functions of the form f(x) = a(b)x, where a > 0, b > 0, and b ≠ 1 have the following characteristics:

Need to Know There are two different shapes of the graphs of an exponential function of the form f(x) = a(b)x, where a > 0, b > 0, and b ≠ 1.

You’re ready! Try the homework from this section. Need to Know To determine if a table of values is an exponential function, look for a pattern in the ordered pairs. As the value of x increases or decreases by a constant amount, the value of y changes by a constant ratio (common factor). You’re ready! Try the homework from this section.