Functions of the Form <em>y = b<sup>x</sup></em>
Functions of the form \(y={b}^{x}\)
Functions of the general form \(y=a{b}^{x}+q\) are called exponential functions. In the equation \(a\) and \(q\) are constants and have different effects on the function.
Example
Question
Complete the following table for each of the functions and draw the graphs on the same system of axes: \(f(x)={2}^{x}\), \(g(x)={3}^{x}\), \(h(x)={5}^{x}\).
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\(f(x)={2}^{x}\) |
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\(g(x)={3}^{x}\) |
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\(h(x)={5}^{x}\) |
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At what point do these graphs intersect?
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Explain why they do not cut the \(x\)-axis.
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Give the domain and range of \(h(x)\).
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As \(x\) increases, does \(h(x)\) increase or decrease?
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Which of these graphs increases at the slowest rate?
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For \(y={k}^{x}\) and \(k>1\), the greater the value of \(k\), the steeper the curve of the graph. True or false?
Complete the following table for each of the functions and draw the graphs on the same system of axes: \(F(x)={(\dfrac{1}{2})}^{x}\), \(G(x)={(\dfrac{1}{3})}^{x}\), \(H(x)={(\dfrac{1}{5})}^{x}\)
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\(F(x)={\left(\cfrac{1}{2}\right)}^{x}\) |
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\(G(x)={\left(\cfrac{1}{3}\right)}^{x}\) |
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\(H(x)={\left(\cfrac{1}{5}\right)}^{x}\) |
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Give the \(y\)-intercept for each function.
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Describe the relationship between the graphs \(f(x)\) and \(F(x)\).
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Describe the relationship between the graphs \(g(x)\) and \(G(x)\).
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Give the domain and range of \(H(x)\).
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For \(y={(\dfrac{1}{k})}^{x}\) and \(k>1\), the greater the value of \(k\), the steeper the curve of the graph. True or false?
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Give the equation of the asymptote for the functions.
Substitute values into the equations
|
\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\(f(x)={2}^{x}\) |
\(\cfrac{1}{4}\) |
\(\cfrac{1}{2}\) |
\(\text{1}\) |
\(\text{2}\) |
\(\text{4}\) |
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\(g(x)={3}^{x}\) |
\(\cfrac{1}{9}\) |
\(\cfrac{1}{3}\) |
\(\text{1}\) |
\(\text{3}\) |
\(\text{9}\) |
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\(h(x)={5}^{x}\) |
\(\cfrac{1}{25}\) |
\(\cfrac{1}{5}\) |
\(\text{1}\) |
\(\text{5}\) |
\(\text{25}\) |
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\(F(x)={\left(\cfrac{1}{2}\right)}^{x}\) |
\(\text{4}\) |
\(\text{2}\) |
\(\text{1}\) |
\(\cfrac{1}{2}\) |
\(\cfrac{1}{4}\) |
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\(G(x)={\left(\cfrac{1}{3}\right)}^{x}\) |
\(\text{9}\) |
\(\text{3}\) |
\(\text{1}\) |
\(\cfrac{1}{3}\) |
\(\cfrac{1}{9}\) |
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\(H(x)={\left(\cfrac{1}{5}\right)}^{x}\) |
\(\text{25}\) |
\(\text{5}\) |
\(\text{1}\) |
\(\cfrac{1}{5}\) |
\(\cfrac{1}{25}\) |
Plot the points and join with a smooth curve
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We notice that all graphs pass through the point \((0;1)\). Any number with exponent \(\text{0}\) is equal to \(\text{1}\).
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The graphs do not cut the \(x\)-axis because you can never get \(\text{0}\) by raising any non-zero number to the power of any other number.
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Domain: \(\{x:x\in \mathbb{R}\}\)
Range: \(\{y:y\in \mathbb{R}, y>0\}\)
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As \(x\) increases, \(h(x)\) increases.
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\(f(x)={2}^{x}\) increases at the slowest rate because it has the smallest base.
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True: the greater the value of \(k\) \((k>1)\), the steeper the graph of \(y={k}^{x}\).
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The \(y\)-intercept is the point \((0;1)\) for all graphs. For any real number \(z\): \({z}^{0}=1 \qquad z \ne 0\).
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\(F(x)\) is the reflection of \(f(x)\) about the \(y\)-axis.
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\(G(x)\) is the reflection of \(g(x)\) about the \(y\)-axis.
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Domain: \(\{x:x\in \mathbb{R}\}\)
Range: \(\{y:y\in \mathbb{R}, y>0\}\)
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True: the greater the value of \(k\) \((k>1)\), the steeper the graph of \(y={\left(\cfrac{1}{k}\right)}^{x}\).
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The equation of the horizontal asymptote is \(y=0\), the \(x\)-axis.
This lesson is part of:
Functions I