Algebra Examples

$f (x) = - 4 \cdot {(0.3)}^{x} + 1$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = - 4 \cdot {(0.3)}^{x} + 1$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 4 \cdot {0.3}^{x} + 1 = 0$ .

$- 4 \cdot {0.3}^{x} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- 4 \cdot {0.3}^{x} = - 1$

Step 1.2.3

Divide each term in $- 4 \cdot {0.3}^{x} = - 1$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- 4 \cdot {0.3}^{x} = - 1$ by $- 4$ .

$\frac{- 4 \cdot {0.3}^{x}}{- 4} = \frac{- 1}{- 4}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 4 \cdot {0.3}^{x}}{- 4} = \frac{- 1}{- 4}$

Step 1.2.3.2.1.2

Divide ${0.3}^{x}$ by $1$ .

${0.3}^{x} = \frac{- 1}{- 4}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Dividing two negative values results in a positive value.

${0.3}^{x} = \frac{1}{4}$

Step 1.2.4

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

$\ln ({0.3}^{x}) = \ln (\frac{1}{4})$

Step 1.2.5

Expand $\ln ({0.3}^{x})$ by moving $x$ outside the logarithm.

$x \ln (0.3) = \ln (\frac{1}{4})$

Step 1.2.6

Divide each term in $x \ln (0.3) = \ln (\frac{1}{4})$ by $\ln (0.3)$ and simplify.

Tap for more steps...

Step 1.2.6.1

Divide each term in $x \ln (0.3) = \ln (\frac{1}{4})$ by $\ln (0.3)$ .

$\frac{x \ln (0.3)}{\ln (0.3)} = \frac{\ln (\frac{1}{4})}{\ln (0.3)}$

Step 1.2.6.2

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.1

Cancel the common factor of $\ln (0.3)$ .

Tap for more steps...

Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{x \ln (0.3)}{\ln (0.3)} = \frac{\ln (\frac{1}{4})}{\ln (0.3)}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{4})}{\ln (0.3)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (\frac{1}{4})}{\ln (0.3)}, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = - 4 \cdot {(0.3)}^{0} + 1$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 4 \cdot {0.3}^{0} + 1$

Step 2.2.2

Remove parentheses.

$y = - 4 \cdot {(0.3)}^{0} + 1$

Step 2.2.3

Simplify $- 4 \cdot {(0.3)}^{0} + 1$ .

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Anything raised to $0$ is $1$ .

$y = - 4 \cdot 1 + 1$

Step 2.2.3.1.2

Multiply $- 4$ by $1$ .

$y = - 4 + 1$

Step 2.2.3.2

Add $- 4$ and $1$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (\frac{1}{4})}{\ln (0.3)}, 0)$

y-intercept(s): $(0, - 3)$

Step 4