What does the graph of $y = b^x$ (where $b > 1$ ) look like?

An increasing curve that passes through (0, 1), approaching the x-axis as x goes to negative infinity.

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

What does the graph of $y = b^x$ (where $b > 1$ ) look like?

An increasing curve that passes through (0, 1), approaching the x-axis as x goes to negative infinity.

What does the graph of $y = \log_b(x)$ (where $b > 1$ ) look like?

An increasing curve that passes through (1, 0), approaching the y-axis as x goes to 0.

How can you identify the base $b$ from the graph of $y = b^x$ ?

Find the point where $x = 1$ . The $y$ -coordinate of that point is the value of $b$ .

How can you identify the base $b$ from the graph of $y = \log_b(x)$ ?

Find the point where $y = 1$ . The $x$ -coordinate of that point is the value of $b$ .

What does the intersection of $y = b^x$ and $y = \log_b(x)$ with $y=x$ represent?

It shows the points where the function and its inverse have the same x and y values.

What are the differences between exponential and logarithmic functions regarding asymptotes?

Exponential: Horizontal asymptote. | Logarithmic: Vertical asymptote.

Compare the domains and ranges of $f(x) = b^x$ and $g(x) = \log_b(x)$ .

Exponential: Domain is $(-\infty, \infty)$ , Range is $(0, \infty)$ . | Logarithmic: Domain is $(0, \infty)$ , Range is $(-\infty, \infty)$ .

Compare the growth rates of exponential and logarithmic functions.

Exponential: Grows rapidly as $x$ increases. | Logarithmic: Grows slowly as $x$ increases.

Compare the behavior of $b^x$ and $\log_b(x)$ as $x$ approaches infinity.

Exponential: Approaches infinity. | Logarithmic: Approaches infinity, but much slower.

Define logarithmic function.

A function of the form $f(x) = a \log_b(x)$ , where $b > 0$ and $b \neq 1$ , and $a \neq 0$ .

Define exponential function.

A function of the form $f(x) = ab^x$ , where $a$ is the coefficient and $b$ is the base.

What is the base of a logarithm?

The value $b$ in $\log_b(x)$ , where $b > 0$ and $b \neq 1$ .

What is the coefficient of an exponential function?

The value $a$ in $f(x) = ab^x$ .

What is the argument of a logarithm?

The input value $x$ in $\log_b(x)$ .

Define the identity function.

The function $h(x) = x$ , a straight line with a slope of 1 passing through the origin.

What is the inverse of an exponential function?

A logarithmic function with the same base.

What is a reflection over the line $y=x$ ?

A transformation where the x and y coordinates of a point are swapped.

What is a horizontal asymptote?

A horizontal line that a graph approaches as $x$ tends to $+\infty$ or $-\infty$ .

What is a vertical asymptote?

A vertical line that a graph approaches as $x$ approaches a certain value.