Question

Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=3^{-x}-2 $$

Short answer

The domain is (-finity, ∞), the range is (-2, ∞), the horizontal asymptote is y=-2, and the y-intercept is (0, -1).

Step-by-step solution

Identify the Parent Function

The parent function is the basic exponential function without any transformations. For this function, we identify the parent function as y=f(x)=3^x.

Reflect Across the Y-Axis

The given function has a negative exponent, which changes the graph of the parent function y=f(x)=3^x . Reflected about the y-axis, the function becomes y=f(x)=3^{-x}.

Vertical Shift Downward

After reflecting the graph of y=f(x)=3^{-x} along the y-axis, shift it downward by 2 units. Therefore, the graph of function f(x)=3^{-x}-2 is the final graph.

Determine the Domain

The domain refers to all the possible input values (x-values). For the function y=f(x)=3^{-x}-2, the domain is all real numbers, represented in interval notation as (-ii, ∞).

Determine the Range

The range refers to all possible output values (y-values). For the function y=f(x)=3^{-x}-2, the range is [-2, ∞).

Identify the Horizontal Asymptote

The horizontal asymptote is the horizontal line that the function approaches as x tends to positive or negative infinity. For the function y=f(x)=3^{-x}-2, the horizontal asymptote is y=-2.

Identify the Y-Intercept

The y-intercept is where the graph intersects the y-axis (where x = 0). Substitute x=0 into the function to get f(0)=3^{-0}-2 ->f(0)=1-2 ->f(0)=-1. So, the y-intercept is y=-1.

Key concepts

Parent Function

The parent function is the simplest form of an exponential function. For the equation we're working with, the parent function is defined as \(y = 3^x\). Think of this as the starting point before any transformations are applied.

The graph of \(y = 3^x\) increases rapidly as x becomes positive and approaches zero as x becomes negative. This is essential because understanding how this base graph behaves helps understand changes when transformations are made.

Transformation

Transformations modify the parent function to create new graphs. For \(f(x) = 3^{-x} - 2\), we apply transformations step-by-step.

First, the negative exponent \(-x\) reflects the graph of \(y=3^x\) across the y-axis. So, the new graph becomes \(y = 3^{-x}\).

Next, the term \(-2\) vertically shifts the graph of \(y = 3^{-x}\) downward by 2 units. The final equation is thus \(f(x) = 3^{-x} - 2\). This vertical shift moves every point on the graph down by 2 units.

Domain and Range

The domain of a function is all the possible x-values for which the function is defined. For \(f(x) = 3^{-x} - 2\), the domain is all real numbers since exponential functions are defined for every x-value. This is written in interval notation as \((-\infty, \infty)\).

The range relates to all possible y-values the function can attain. After shifting the graph downward by 2 units, the lowest output value (y-value) becomes \(-2\). Therefore, the range of \(f(x)\) is \([-2, \infty)\). This means the function can attain any y-value greater than or equal to \(-2\).

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph approaches but never touches as \(x\) tends to positive or negative infinity. For the function \(f(x) = 3^{-x} - 2\), the graph comes arbitrarily close to, but doesn’t reach, the line \(y = -2\) as \(x\) becomes infinitely large or small.

This means the graph will level out at and tend towards \(y = -2\) without ever crossing this line, making \(y = -2\) the horizontal asymptote for this function.

Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept for \(f(x) = 3^{-x} - 2\), substitute \(x = 0\) into the equation:

\[ f(0) = 3^{-0} - 2 = 1 - 2 = -1 \]

Therefore, the y-intercept is at the point \(y = -1\).