What does the graph of an increasing exponential function tell us?

It indicates exponential growth, where the rate of increase accelerates over time.

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What does the graph of an increasing exponential function tell us?

It indicates exponential growth, where the rate of increase accelerates over time.

What does the graph of a decreasing exponential function tell us?

It indicates exponential decay, where the rate of decrease slows down over time.

What does the graph of a logarithmic function tell us?

It shows a slow rate of increase, approaching a vertical asymptote.

How does the base of an exponential function affect the graph?

A larger base results in faster growth or decay.

How does the base of a logarithmic function affect the graph?

The base affects the steepness and position of the graph relative to the y-axis.

What does a semi-log plot of exponential data look like?

A straight line, indicating a constant rate of growth or decay on a logarithmic scale.

How can you determine if a graph represents an exponential function?

Look for rapid growth or decay and a horizontal asymptote.

How can you determine if a graph represents a logarithmic function?

Look for a slow rate of increase and a vertical asymptote.

What does the y-intercept of an exponential function represent?

The initial value of the function.

What does the x-intercept of a logarithmic function represent?

The value for which the argument of the logarithm is equal to 1.

How do you solve an exponential equation?

Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.

How do you solve a logarithmic equation?

Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.

How do you find the inverse of a function?

Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$ .

How do you evaluate a composite function?

Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.

How do you determine if two functions are inverses?

Show that $f(g(x)) = x$ and $g(f(x)) = x$ .

How do you graph an exponential function?

Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.

How do you graph a logarithmic function?

Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.

How do you use properties of logarithms to simplify expressions?

Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.

How to determine the equation of an exponential function from data points?

Start with $f(x) = ab^x$ . 2. Use two data points to create a system of equations. 3. Solve for a and b.

How do you solve for the half-life in an exponential decay problem?

Set up the equation $0.5 = b^t$ , where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.

What is the formula for the nth term of an arithmetic sequence?

$u_n = a + (n-1)d$ , where 'a' is the first term and 'd' is the common difference.

What is the formula for the nth term of a geometric sequence?

$u_n = ar^{n-1}$ , where 'a' is the first term and 'r' is the common ratio.

What is the general form of an exponential function?

$f(x) = ab^x$

How do you denote the inverse of a function f(x)?

$f^{-1}(x)$

What is the relationship between a function and its inverse?

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

What is the formula for a composite function of f and g?

$(f \circ g)(x) = f(g(x))$

How to find the inverse of a function y=f(x)?

Swap x and y, then solve for y.

What is the logarithmic form of $b^y = x$ ?

$y = \log_b(x)$

What is the exponential form of $y = \log_b(x)$ ?

$b^y = x$

What is the change of base formula for logarithms?

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

All Flashcards

What does the graph of an increasing exponential function tell us?

It indicates exponential growth, where the rate of increase accelerates over time.

What does the graph of a decreasing exponential function tell us?

It indicates exponential decay, where the rate of decrease slows down over time.

What does the graph of a logarithmic function tell us?

It shows a slow rate of increase, approaching a vertical asymptote.

How does the base of an exponential function affect the graph?

A larger base results in faster growth or decay.

How does the base of a logarithmic function affect the graph?

The base affects the steepness and position of the graph relative to the y-axis.

What does a semi-log plot of exponential data look like?

A straight line, indicating a constant rate of growth or decay on a logarithmic scale.

How can you determine if a graph represents an exponential function?

Look for rapid growth or decay and a horizontal asymptote.

How can you determine if a graph represents a logarithmic function?

Look for a slow rate of increase and a vertical asymptote.

What does the y-intercept of an exponential function represent?

The initial value of the function.

What does the x-intercept of a logarithmic function represent?

The value for which the argument of the logarithm is equal to 1.

How do you solve an exponential equation?

Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.

How do you solve a logarithmic equation?

Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.

How do you find the inverse of a function?

Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$ .

How do you evaluate a composite function?

Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.

How do you determine if two functions are inverses?

Show that $f(g(x)) = x$ and $g(f(x)) = x$ .

How do you graph an exponential function?

Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.

How do you graph a logarithmic function?

Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.

How do you use properties of logarithms to simplify expressions?

Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.

How to determine the equation of an exponential function from data points?

Start with $f(x) = ab^x$ . 2. Use two data points to create a system of equations. 3. Solve for a and b.

How do you solve for the half-life in an exponential decay problem?

Set up the equation $0.5 = b^t$ , where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.

What is the formula for the nth term of an arithmetic sequence?

$u_n = a + (n-1)d$ , where 'a' is the first term and 'd' is the common difference.

What is the formula for the nth term of a geometric sequence?

$u_n = ar^{n-1}$ , where 'a' is the first term and 'r' is the common ratio.

What is the general form of an exponential function?

$f(x) = ab^x$

How do you denote the inverse of a function f(x)?

$f^{-1}(x)$

What is the relationship between a function and its inverse?

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

What is the formula for a composite function of f and g?

$(f \circ g)(x) = f(g(x))$

How to find the inverse of a function y=f(x)?

Swap x and y, then solve for y.

What is the logarithmic form of $b^y = x$ ?

$y = \log_b(x)$

What is the exponential form of $y = \log_b(x)$ ?

$b^y = x$

What is the change of base formula for logarithms?

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$