Finite Math Examples

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

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 $x$ .

$0 = 4 \cdot {(\frac{1}{2})}^{x - 2} - 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 \cdot {(\frac{1}{2})}^{x - 2} - 2 = 0$ .

$4 \cdot {(\frac{1}{2})}^{x - 2} - 2 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{2}$ .

$4 \cdot \frac{1^{x - 2}}{2^{x - 2}} - 2 = 0$

Step 1.2.2.2

One to any power is one.

$4 \cdot \frac{1}{2^{x - 2}} - 2 = 0$

Step 1.2.2.3

Combine $4$ and $\frac{1}{2^{x - 2}}$ .

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

Step 1.2.3

Add $2$ to both sides of the equation.

$\frac{4}{2^{x - 2}} = 2$

Step 1.2.4

Move $2^{x - 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

$2^{2 - (x - 2)} = 2$

Step 1.2.7

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

$2^{2 - (x - 2)} = 2^{1}$

Step 1.2.8

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

$2 - (x - 2) = 1$

Step 1.2.9

Solve for $x$ .

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Step 1.2.9.1

Simplify $2 - (x - 2)$ .

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Step 1.2.9.1.1

Simplify each term.

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Step 1.2.9.1.1.1

Apply the distributive property.

$2 - x - - 2 = 1$

Step 1.2.9.1.1.2

Multiply $- 1$ by $- 2$ .

$2 - x + 2 = 1$

Step 1.2.9.1.2

Add $2$ and $2$ .

$- x + 4 = 1$

Step 1.2.9.2

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

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Step 1.2.9.2.1

Subtract $4$ from both sides of the equation.

$- x = 1 - 4$

Step 1.2.9.2.2

Subtract $4$ from $1$ .

$- x = - 3$

Step 1.2.9.3

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 1.2.9.3.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 1.2.9.3.2

Simplify the left side.

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Step 1.2.9.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 1.2.9.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 1.2.9.3.3

Simplify the right side.

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Step 1.2.9.3.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 4 \cdot {(\frac{1}{2})}^{(0) - 2} - 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot {(\frac{1}{2})}^{0 - 2} - 2$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot {(\frac{1}{2})}^{(0) - 2} - 2$

Step 2.2.3

Simplify $4 \cdot {(\frac{1}{2})}^{(0) - 2} - 2$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $2$ from $0$ .

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

Step 2.2.3.1.2

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 2.2.3.1.3

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

$y = 4 \cdot 4 - 2$

Step 2.2.3.1.4

Multiply $4$ by $4$ .

$y = 16 - 2$

Step 2.2.3.2

Subtract $2$ from $16$ .

$y = 14$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 14)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 14)$

Step 4