Algebra Examples

$f (x) = 5^{x - 5} - 5$

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 $x$ .

$0 = 5^{x - 5} - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5^{x - 5} - 5 = 0$ .

$5^{x - 5} - 5 = 0$

Step 1.2.2

Add $5$ to both sides of the equation.

$5^{x - 5} = 5$

Step 1.2.3

Create equivalent expressions in the equation that all have equal bases.

$5^{x - 5} = 5^{1}$

Step 1.2.4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x - 5 = 1$

Step 1.2.5

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.1

Add $5$ to both sides of the equation.

$x = 1 + 5$

Step 1.2.5.2

Add $1$ and $5$ .

$x = 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0)$

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 $y$ .

$y = 5^{(0) - 5} - 5$

Step 2.2

Simplify $5^{(0) - 5} - 5$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $5$ from $0$ .

$y = 5^{- 5} - 5$

Step 2.2.1.2

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

$y = \frac{1}{5^{5}} - 5$

Step 2.2.1.3

Raise $5$ to the power of $5$ .

$y = \frac{1}{3125} - 5$

Step 2.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3125}{3125}$ .

$y = \frac{1}{3125} - 5 \cdot \frac{3125}{3125}$

Step 2.2.3

Combine $- 5$ and $\frac{3125}{3125}$ .

$y = \frac{1}{3125} + \frac{- 5 \cdot 3125}{3125}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 5 \cdot 3125}{3125}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 5$ by $3125$ .

$y = \frac{1 - 15625}{3125}$

Step 2.2.5.2

Subtract $15625$ from $1$ .

$y = \frac{- 15624}{3125}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{15624}{3125}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{15624}{3125})$

Step 3

List the intersections.

x-intercept(s): $(6, 0)$

y-intercept(s): $(0, - \frac{15624}{3125})$

Step 4