Precalculus Examples

$y = 7 (2^{4 x - 10}) - 28$

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 $x$ .

$0 = 7 (2^{4 x - 10}) - 28$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $7 \cdot 2^{4 x - 10} - 28 = 0$ .

$7 \cdot 2^{4 x - 10} - 28 = 0$

Step 1.2.2

Add $28$ to both sides of the equation.

$7 \cdot 2^{4 x - 10} = 28$

Step 1.2.3

Divide each term in $7 \cdot 2^{4 x - 10} = 28$ by $7$ and simplify.

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Step 1.2.3.1

Divide each term in $7 \cdot 2^{4 x - 10} = 28$ by $7$ .

$\frac{7 \cdot 2^{4 x - 10}}{7} = \frac{28}{7}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $7$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{7 \cdot 2^{4 x - 10}}{7} = \frac{28}{7}$

Step 1.2.3.2.1.2

Divide $2^{4 x - 10}$ by $1$ .

$2^{4 x - 10} = \frac{28}{7}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $28$ by $7$ .

$2^{4 x - 10} = 4$

Step 1.2.4

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

$2^{4 x - 10} = 2^{2}$

Step 1.2.5

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

$4 x - 10 = 2$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

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

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Step 1.2.6.1.1

Add $10$ to both sides of the equation.

$4 x = 2 + 10$

Step 1.2.6.1.2

Add $2$ and $10$ .

$4 x = 12$

Step 1.2.6.2

Divide each term in $4 x = 12$ by $4$ and simplify.

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Step 1.2.6.2.1

Divide each term in $4 x = 12$ by $4$ .

$\frac{4 x}{4} = \frac{12}{4}$

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{12}{4}$

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{4}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Divide $12$ by $4$ .

$x = 3$

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 $y$ .

$y = 7 (2^{4 (0) - 10}) - 28$

Step 2.2

Solve the equation.

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Step 2.2.1

Multiply $7$ by $2^{4 (0) - 10}$ .

$y = 7 \cdot 2^{4 (0) - 10} - 28$

Step 2.2.2

Remove parentheses.

$y = 7 (2^{4 (0) - 10}) - 28$

Step 2.2.3

Simplify $7 (2^{4 (0) - 10}) - 28$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $4$ by $0$ .

$y = 7 \cdot 2^{0 - 10} - 28$

Step 2.2.3.1.2

Subtract $10$ from $0$ .

$y = 7 \cdot 2^{- 10} - 28$

Step 2.2.3.1.3

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

$y = 7 \cdot \frac{1}{2^{10}} - 28$

Step 2.2.3.1.4

Raise $2$ to the power of $10$ .

$y = 7 \cdot \frac{1}{1024} - 28$

Step 2.2.3.1.5

Combine $7$ and $\frac{1}{1024}$ .

$y = \frac{7}{1024} - 28$

Step 2.2.3.2

To write $- 28$ as a fraction with a common denominator, multiply by $\frac{1024}{1024}$ .

$y = \frac{7}{1024} - 28 \cdot \frac{1024}{1024}$

Step 2.2.3.3

Combine $- 28$ and $\frac{1024}{1024}$ .

$y = \frac{7}{1024} + \frac{- 28 \cdot 1024}{1024}$

Step 2.2.3.4

Combine the numerators over the common denominator.

$y = \frac{7 - 28 \cdot 1024}{1024}$

Step 2.2.3.5

Simplify the numerator.

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Step 2.2.3.5.1

Multiply $- 28$ by $1024$ .

$y = \frac{7 - 28672}{1024}$

Step 2.2.3.5.2

Subtract $28672$ from $7$ .

$y = \frac{- 28665}{1024}$

Step 2.2.3.6

Move the negative in front of the fraction.

$y = - \frac{28665}{1024}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{28665}{1024})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{28665}{1024})$

Step 4