Question

As the base \(a\) of an exponential function \(f(x)=a^{x}\), where \(a>1\), increases, what happens to its graph for \(x>0 ?\) What happens to its graph for \(x<0 ?\)

Short answer

For \(x > 0\), the graph rises faster. For \(x < 0\), the graph descends more quickly.

Step-by-step solution

Understanding the Exponential Function

An exponential function is of the form \(f(x) = a^x\), where the base \(a\) is greater than 1. Examine how changes in \(a\) affect the graph of the function.

Analyze the Behavior for \(x > 0\)

For values of \(x > 0\), increasing the base \(a\) results in the graph of the function growing steeper. This occurs because larger bases magnify the effect of exponentiation, causing the function to increase more rapidly.

Analyze the Behavior for \(x < 0\)

For values of \(x < 0\), increasing the base \(a\) makes the graph decrease more steeply. This is because raising larger bases to negative exponents results in smaller values approaching zero faster.

Summary

In summary, as the base \(a\) increases: - For \(x > 0\), the graph becomes steeper and rises more quickly.- For \(x < 0\), the graph descends more sharply and approaches zero faster.

Key concepts

Effects of Changing the Base

In an exponential function of the form \(f(x) = a^x\), the base \(a\) plays a crucial role in determining the graph's behavior. When \(a > 1\), changing the value of \(a\) can increase or decrease the steepness of the graph.

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• When the base \(a\) increases, the graph of the function changes significantly.
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• For \(x > 0\) (positive exponents), the graph rises faster as \(a\) increases.
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• For \(x < 0\) (negative exponents), the graph falls more steeply and approaches zero quicker as \(a\) increases.
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\ Why does this happen? The reason lies in the nature of exponentiation: When the base is larger, each step of exponentiation has a more pronounced effect, either increasing or decreasing the function's value more rapidly depending on the exponent's sign.

Graph Behavior for Positive Exponents

For an exponential function \(f(x) = a^x\) with \(a > 1\) and \(x > 0\), an increase in the base \(a\) results in a steeper graph.

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• As \(a\) becomes larger, say from 2 to 3, the rate at which \(f(x)\) increases becomes faster.
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• For any given \(x > 0\), a larger base means the value of \(f(x) = a^x\) becomes significantly larger.
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\ An easy way to visualize this is by considering small changes first. If the base changes from 2 to 3, for \(x = 1\), \(f(1)\) changes from 2 to 3. For \(x = 2\), \(f(2)\) changes from 4 to 9, showing a quicker increase.

In essence, the graph starts to 'shoot up' quicker, depicting a steeper slope as the base \(a\) increases for positive \(x\) values.

Graph Behavior for Negative Exponents

For an exponential function \(f(x) = a^x\) where \(a > 1\) and \(x < 0\), increasing the base \(a\) makes the graph fall more sharply.

Here’s a deeper look:

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• When \(x\) is negative, \(a^x\) translates into \(1/(a^|x|)\), where \( |x| \) is the absolute value of \(x\).
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• An increase in \(a\) then means \(1/(a^|x|)\) becomes smaller more quickly.
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\ To visualize this, consider base changes. If \(a\) increases from 2 to 3:

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• For \(x = -1\), \(f(-1)\) changes from 1/2 to 1/3.
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• For \(x = -2\), \(f(-2)\) changes from 1/4 to 1/9.
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As \(a\) increases, the graph gets closer to zero faster, indicating a steeper descent. For negative exponents, each jump in base size makes the function's value decrease more rapidly, thus causing the graph to 'dive' quicker as \(a\) becomes larger.