Mathematics
Grade 11
15 min
Identify linear and exponential functions
Identify linear and exponential functions
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1
Introduction & Learning Objectives
Learning Objectives
Differentiate between linear and exponential functions based on their general algebraic forms.
Identify a linear function from a table of values by calculating a constant common difference.
Identify an exponential function from a table of values by calculating a constant common ratio.
Distinguish between the graphical representations of linear and exponential functions.
Analyze a word problem to determine if the relationship it describes is linear or exponential.
Write the equation for a linear or exponential function given a table of values or a description of the relationship.
Ever wonder how your savings can grow slowly at first, then suddenly skyrocket? 🚀 Let's explore the powerful mathematics that distinguishes steady growth from explosive grow...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function that has a constant rate of change. For every constant change in the input (x), there is a constant change in the output (y) by addition or subtraction. Its graph is a straight line.A taxi fare that costs a flat fee of $3 plus $2 per mile can be modeled by the linear function f(x) = 2x + 3.
Exponential FunctionA function in which a constant change in the input (x) results in the output (y) being multiplied by a constant factor. Its graph is a curve that increases or decreases at an accelerating rate.A bacterial culture that doubles every hour, starting with 100 bacteria, can be modeled by the exponential function f(x) = 100(2)^x.
Common DifferenceThe constant value that is added to each output to get the next output in a linear pattern. It i...
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Core Formulas
Linear Function General Form
f(x) = mx + b
Use this form to model relationships with a constant rate of change. 'm' represents the slope (the common difference when Δx=1), and 'b' represents the initial value or y-intercept.
Exponential Function General Form
f(x) = a \cdot b^x
Use this form to model relationships with a constant multiplicative factor. 'a' is the initial value (where a ≠0), and 'b' is the common ratio or growth/decay factor (where b > 0 and b ≠1).
Test for Linearity from a Table
\frac{y_2 - y_1}{x_2 - x_1} = \text{constant}
For any two points in a table, the rate of change (slope) must be constant. If the x-values increase by a regular interval, you can simply check if the difference between consecutive y-v...
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Challenging
The table below represents a linear function, but one y-value is missing. What is the value of 'k'?
x | 1 | 2 | 3 | 4
--|---|---|---|---
y | 4 | 9 | k | 19
A.13
B.14
C.15
D.16
Challenging
The table below represents an exponential function, but one y-value is missing. What is the value of 'k'?
x | 0 | 1 | 2 | 3
--|---|---|---|---
y | 4 | 6 | k | 13.5
A.9
B.8
C.10
D.11.5
Challenging
Function A is linear: f(x) = 50x + 500. Function B is exponential: g(x) = 500(1.1)^x. Both model the value of an investment. Which statement accurately compares them for x > 0?
A.Function A will always have a greater value than Function B.
B.Function B will always have a greater value than Function A.
C.Function A initially grows faster, but Function B will eventually have a much greater value.
D.Function B initially grows faster, but Function A will eventually have a much greater value.
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