Algebra Examples

$f (x) = - 2 \cdot 4^{x - 2}$

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 $x$ .

$0 = - 2 \cdot 4^{x - 2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 \cdot 4^{x - 2} = 0$ .

$- 2 \cdot 4^{x - 2} = 0$

Step 1.2.2

Multiply $- 2 \cdot 4^{x - 2}$ .

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Step 1.2.2.1

Factor out negative.

$- (2 \cdot 4^{x - 2}) = 0$

Step 1.2.2.2

Rewrite $4$ as $2^{2}$ .

$- (2 \cdot {(2^{2})}^{x - 2}) = 0$

Step 1.2.2.3

Multiply the exponents in ${(2^{2})}^{x - 2}$ .

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Step 1.2.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- (2 \cdot 2^{2 (x - 2)}) = 0$

Step 1.2.2.3.2

Apply the distributive property.

$- (2 \cdot 2^{2 x + 2 \cdot - 2}) = 0$

Step 1.2.2.3.3

Multiply $2$ by $- 2$ .

$- (2 \cdot 2^{2 x - 4}) = 0$

Step 1.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2^{1 + 2 x - 4} = 0$

Step 1.2.2.5

Subtract $4$ from $1$ .

$- 2^{2 x - 3} = 0$

Step 1.2.3

Divide each term in $- 2^{2 x - 3} = 0$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- 2^{2 x - 3} = 0$ by $- 1$ .

$\frac{- 2^{2 x - 3}}{- 1} = \frac{0}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{2^{2 x - 3}}{1} = \frac{0}{- 1}$

Step 1.2.3.2.2

Divide $2^{2 x - 3}$ by $1$ .

$2^{2 x - 3} = \frac{0}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $- 1$ .

$2^{2 x - 3} = 0$

Step 1.2.4

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

$\ln (2^{2 x - 3}) = \ln (0)$

Step 1.2.5

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

Undefined

Step 1.2.6

There is no solution for $2^{2 x - 3} = 0$

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 $y$ .

$y = - 2 \cdot 4^{(0) - 2}$

Step 2.2

Simplify $- 2 \cdot 4^{(0) - 2}$ .

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Step 2.2.1

Subtract $2$ from $0$ .

$y = - 2 \cdot 4^{- 2}$

Step 2.2.2

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

$y = - 2 \cdot \frac{1}{4^{2}}$

Step 2.2.3

Raise $4$ to the power of $2$ .

$y = - 2 \cdot \frac{1}{16}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \cdot \frac{1}{16}$

Step 2.2.4.2

Factor $2$ out of $16$ .

$y = 2 \cdot - 1 \cdot \frac{1}{2 \cdot 8}$

Step 2.2.4.3

Cancel the common factor.

$y = 2 \cdot - 1 \cdot \frac{1}{2 \cdot 8}$

Step 2.2.4.4

Rewrite the expression.

$y = - 1 \cdot \frac{1}{8}$

Step 2.2.5

Rewrite $- 1 (\frac{1}{8})$ as $- (\frac{1}{8})$ .

$y = - \frac{1}{8}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{8})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{1}{8})$

Step 4