Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{4} \cdot e^{x - 3} = 0$ .

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

Step 1.2.2

Multiply both sides of the equation by $- 4$ .

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

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $- 4 (- \frac{1}{4} \cdot e^{x - 3})$ .

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Step 1.2.3.1.1.1

Combine $e^{x - 3}$ and $\frac{1}{4}$ .

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

Step 1.2.3.1.1.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.2.1

Move the leading negative in $- \frac{e^{x - 3}}{4}$ into the numerator.

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

Step 1.2.3.1.1.2.2

Factor $4$ out of $- 4$ .

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

Step 1.2.3.1.1.2.3

Cancel the common factor.

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

Step 1.2.3.1.1.2.4

Rewrite the expression.

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

Step 1.2.3.1.1.3

Multiply.

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Step 1.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.3.1.1.3.2

Multiply $e^{x - 3}$ by $1$ .

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 4$ by $0$ .

$e^{x - 3} = 0$

Step 1.2.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 1.2.5

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 1.2.6

There is no solution for $e^{x - 3} = 0$

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Simplify $- \frac{1}{4} \cdot e^{(0) - 3}$ .

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Step 2.2.1

Subtract $3$ from $0$ .

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

Step 2.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2.3

Multiply $\frac{1}{e^{3}}$ by $\frac{1}{4}$ .

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

Step 2.2.4

Move $4$ to the left of $e^{3}$ .

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

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $None$

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

Step 4