Find the x and y Intercepts f(x)=-1/4e^(x-3)
The question is asking for the identification of the points on the coordinate plane where the graph of the given function f(x) = -1/4 * e^(x-3) crosses the x-axis and y-axis. The x-intercept(s) refers to the point(s) where the function has a value of zero (where the graph crosses the x-axis), and the y-intercept is the point where the graph of the function crosses the y-axis, which occurs when the value of x is zero.
$f \left(\right. x \left.\right) = - \frac{1}{4} e^{x - 3}$
Answer
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:
x-intercepts: Points where the graph of a function crosses the x-axis. To find them, set $y=0$ and solve for $x$.
y-intercepts: Points where the graph of a function crosses the y-axis. To find them, set $x=0$ and solve for $y$.
Exponential functions: Functions of the form $f(x) = a \cdot e^{bx+c}$, where $e$ is the base of the natural logarithm.
Natural logarithm: The logarithm to the base $e$, denoted as $\ln(x)$. It is the inverse operation to taking the power of $e$.
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.
Negative exponent rule: For any nonzero number $b$ and positive integer $n$, $b^{-n} = \frac{1}{b^n}$.
Solving exponential equations: To solve equations with the variable in the exponent, logarithms are often used to isolate the variable.
