Find the x and y Intercepts f(x)=-1/4e^(x-3)

Solution:

Step 1: Determine the x-intercepts

Step 1.1: Set $y$ to $0$ and solve for $x$

$$0=-\frac{1}{4}{e}^{x-3}$$

Step 1.2: Proceed to solve the equation

Step 1.2.1: Restate the equation

$$-\frac{1}{4}{e}^{x-3}=0$$

Step 1.2.2: Multiply by the reciprocal of $-\frac{1}{4}$

$$-4(-\frac{1}{4}{e}^{x-3})=-4(0)$$

Step 1.2.3: Simplify the equation

Step 1.2.3.1: Focus on the left side

Step 1.2.3.1.1: Apply multiplication

$$-4(-\frac{e^{x-3}}{4})=0$$

Step 1.2.3.1.1.2: Remove the common factor

$$e^{x-3}=0$$

Step 1.2.3.2: Simplify the right side

$$e^{x-3}=0$$

Step 1.2.4: Logarithmic approach to isolate $x$

$$\ln (e^{x-3})=\ln (0)$$

Step 1.2.5: Recognize the impossibility of the solution

Undefined

Step 1.2.6: Conclude no x-intercepts exist

No solution

Step 1.3: State the x-intercepts

x-intercepts: None

Step 2: Identify the y-intercepts

Step 2.1: Set $x$ to $0$ and solve for $y$

$$y=-\frac{1}{4}{e}^{0-3}$$

Step 2.2: Simplify the expression

Step 2.2.1: Calculate the exponent

$$y=-\frac{1}{4}{e}^{-3}$$

Step 2.2.2: Apply the negative exponent rule

$$y=-\frac{1}{4}\cdot \frac{1}{e^3}$$

Step 2.2.3: Combine the fractions

$$y=-\frac{1}{4e^3}$$

Step 2.3: Express the y-intercepts in point form

y-intercepts: $(0,-\frac{1}{4e^3})$

Step 3: Compile the intercepts

x-intercepts: None y-intercepts: $(0,-\frac{1}{4e^3})$

Knowledge Notes:

1. x-intercepts: Points where the graph of a function crosses the x-axis. To find them, set $y=0$ and solve for $x$.

2. y-intercepts: Points where the graph of a function crosses the y-axis. To find them, set $x=0$ and solve for $y$.

3. Exponential functions: Functions of the form $f(x)=a\cdot e^{bx+c}$, where $e$ is the base of the natural logarithm.

4. Natural logarithm: The logarithm to the base $e$, denoted as $\ln (x)$. It is the inverse operation to taking the power of $e$.

5. Undefined logarithms: The natural logarithm of zero, $\ln (0)$, is undefined because there is no number that $e$ can be raised to in order to get zero.

6. Negative exponent rule: For any nonzero number $b$ and positive integer $n$, $b^{-n}=\frac{1}{b^n}$.

7. Solving exponential equations: To solve equations with the variable in the exponent, logarithms are often used to isolate the variable.