Question

y = 0.4^x
(a) choose the graph that best represents the shape of this function.
(graphs are shown here, omitted as per instruction)
(b) the values of y are ______ for all values of x.
options: positive, negative
(c) find the domain and range. write the answers as inequalities, using x or y as appropriate.
domain: ______
range: ______
(d) give the x - intercept(s) and y - intercept(s). write
one\ if applicable.
x - intercept(s): ______
y - intercept(s): ______
(e) choose the best description for the end behavior.
options:
as x increases, y increases. as x decreases, y approaches 0.
as x increases, y approaches 0. as x decreases, y increases.
as x increases, y decreases. as x decreases, y approaches 0.
as x increases, y approaches 0. as x decreases, y decreases.

Explanation:

Step1: Analyze the exponential function

The function is $y=0.4^x$, where $0<0.4<1$, so it is a decreasing exponential function that approaches 0 as $x$ increases, and grows as $x$ decreases, staying positive for all $x$.

Step2: Match to correct graph

A decreasing exponential function starting high on the left, approaching the horizontal asymptote $y=0$ on the right, which is Graph 2.

Step3: Determine sign of y-values

Any positive number raised to any real power is positive, so $y>0$ for all $x$.

Step4: Find domain of the function

Exponential functions accept all real numbers as inputs, so domain is all real $x$.

Step5: Find range of the function

Since $0.4^x$ is always positive and approaches 0 (never reaching it), range is $y>0$.

Step6: Calculate x-intercept

x-intercept occurs where $y=0$. $0.4^x=0$ has no solution, so x-intercept is None.

Step7: Calculate y-intercept

y-intercept occurs where $x=0$. $y=0.4^0=1$, so y-intercept is $(0,1)$.

Step8: Describe end behavior

For $y=0.4^x$ (base between 0 and 1): as $x$ increases, $y$ approaches 0; as $x$ decreases, $y$ increases.

Answer:

(a) Graph 2
(b) positive
(c) Domain: $-\infty < x < \infty$
Range: $y > 0$
(d) x-intercept(s): None
y-intercept(s): $(0, 1)$
(e) As x increases, y approaches 0. As x decreases, y increases.