We will analyze the graphs of exponential, quadratic, and linear functions. By doing this we can then describe the portions of the graph where the functions are , , , or , as well as , , , and .
Exponential Functions, f(x)=ab^x+c
First let's analyze the graph of an , y=ab^x, where a≠0, b>0, and b≠1.
Note that depending on the values of the a, b, and c, there are different possibilities for the properties we are going to describe. Now, let's analyze the behavior of y=ab^x+c, where a>0, b>1, and c≥ 0.
In this case there is one , no , and the function is increasing and positive in the entire . We see that there are no relative maxima or minima, there are no symmetries, y approaches c when x decreases, and y increases to infinity when x increases.
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Try the interactive diagram to see this!
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If we analyze other possibilities for a, b, and c, we will find the following diagrams.
Now, let's write down their properties in the following table.
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Exponential Functions
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| Coefficients
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a>0, b>1, c≥ 0
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a>0, b>1, c<0
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a>0, 0
| a>0, 0
| a<0, b>1, c<0
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a<0, b>1, c≥ 0
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a<0, 0
| a<0, 0 |
| y-intercepts
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One
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One
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One
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One
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One
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One
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One
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One
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| x-intercepts
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None
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One
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None
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One
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None
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One
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None
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One
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| Increasing/ Decreasing
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Increasing over the entire domain.
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Increasing over the entire domain.
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Decreasing over the entire domain.
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Decreasing over the entire domain.
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Decreasing over the entire domain.
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Decreasing over the entire domai
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Increasing over the entire domai
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Increasing over the entire domai
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| Positive/ Negative
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Positive over the entire domai
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Negative on the left of the x-intercept, positive on the righ
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Positive over the entire domai
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Positive on the left of the x-intercept, negative on the righ
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Negative over the entire domai
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Positive on the left of the x-intercept, negative on the righ
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Negative over the entire domai
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Negative on the left of the x-intercept, positive on the righ
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| Relative Minima and Maxima
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None
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None
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None
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None
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None
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None
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None
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None
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| Symmetries
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None
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None
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None
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None
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None
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None
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None
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None
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| End Behavior
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y approaches c when x decreases, y approaches ∞ when x increase
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y approaches c when x decreases, y approaches ∞ when x increase
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y approaches ∞ when x decreases, y approaches c when x increase
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y approaches ∞ when x decreases, y approaches c when x increase
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y approaches c when x decreases, y approaches negative -∞ when x increase
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y approaches c when x decreases, y approaches negative -∞ when x increase
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y approaches -∞ when x decreases, y approaches c when x increase
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y approaches -∞ when x decreases, y approaches c when x increase
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Quadratic Functions, g(x)=ax^2+c
Let's analyze the graph of a , y=ax^2+c.
If we analyze different values of parameters a and c, we find the following different types of graphs of quadratic functions.
Let's write down their properties in the following table.
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Quadratic Function
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| Conditions
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a>0, c>0
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a>0, c=0
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a>0, c<0
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a<0, c<0
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a<0, c=0
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a<0, c>0
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| y-intercepts
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One
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One
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One
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One
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One
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One
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| x-intercepts
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None
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One
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Two
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None
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One
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Two
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| Increasing/ Decreasing
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Decreasing on the left of the vertex, increasing on the right
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Decreasing on the left of the vertex, increasing on the right
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Decreasing on the left of the vertex, increasing on the right
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Increasing on the left of the vertex, decreasing on the right
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Increasing on the left of the vertex, decreasing on the right
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Increasing on the left of the vertex, decreasing on the right
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| Positive / Negative
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Positive over the entire domain
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Positive on the left and on the right of the vertex
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Positive on the left of the first zero and on the right of the second zero, negative between the zeros
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Negative over the entire domain
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Negative on the left and on the right of the vertex
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Negative on the left of the first zero and on the right of the second zero, positive between the zeros
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| Relative Minima and Maxima
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One relative minimum at the vertex
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One relative minimum at the vertex
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One relative minimum at the vertex
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One relative maximum at the vertex
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One relative maximum at the vertex
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One relative maximum at the vertex
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| Symmetries
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Line symmetry about the vertical line through the vertex
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Line symmetry about the vertical line through the vertex
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Line symmetry about the vertical line through the vertex
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Line symmetry about the vertical line through the vertex
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Line symmetry about the vertical line through the vertex
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Line symmetry about the vertical line through the vertex
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| End Behavior
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y increases to ∞ when x decreases and when x increases
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y increases to ∞ when x decreases and when x increases
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y increases to ∞ when x decreases and when x increases
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y decreases to -∞ when x decreases and when x increases
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y decreases to -∞ when x decreases and when x increases
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y decreases to -∞ when x decreases and when x increases
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Linear Functions, h(x)=ax+c
Finally we will analyze the graph of a , y=ax+c.
If we analyze different values of parameters a and c, we find the following different types of graphs of linear functions.
Now, let's write down their properties.
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Linear Functions
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| Coefficients
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a<0
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a>0
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| y-intercepts
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One
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One
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| x-intercepts
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One
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One
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| Increasing/ Decreasing
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Increasing over the entire domain
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Decreasing over the entire domain
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| Positive/ Negative
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Negative on the left of the zero, positive on the right
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Positive on the left of the zero, negative on the right
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| Relative Minima and Maxima
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None
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None
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| Symmetries
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Point symmetry about any point on the line
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Point symmetry about any point on the line
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| End Behavior
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y decreases to -∞ when x decreases, y increases to ∞ when x increases
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y increases to ∞ when x decreases, y decreases to -∞ when x increases
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Question
We are asked which function eventually exceeds the others. Analyzing the graphs of exponential, quadratic, and linear functions, we can determine that exponential functions grow the fastest.
