Precalculus Examples

$f (x) = 3 {(x - 2)}^{2} - 3$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Simplify $3 {(x - 2)}^{2} - 3$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$0 = 3 ((x - 2) (x - 2)) - 3$

Step 1.2.2.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.2.2.1.2.1

Apply the distributive property.

$0 = 3 (x (x - 2) - 2 (x - 2)) - 3$

Step 1.2.2.1.2.2

Apply the distributive property.

$0 = 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 3$

Step 1.2.2.1.2.3

Apply the distributive property.

$0 = 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 3$

Step 1.2.2.1.3

Simplify and combine like terms.

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Step 1.2.2.1.3.1

Simplify each term.

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Step 1.2.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.2.2.1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.2.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$0 = 3 (x^{2} - 2 x - 2 x + 4) - 3$

Step 1.2.2.1.3.2

Subtract $2 x$ from $- 2 x$ .

$0 = 3 (x^{2} - 4 x + 4) - 3$

Step 1.2.2.1.4

Apply the distributive property.

$0 = 3 x^{2} + 3 (- 4 x) + 3 \cdot 4 - 3$

Step 1.2.2.1.5

Simplify.

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Step 1.2.2.1.5.1

Multiply $- 4$ by $3$ .

$0 = 3 x^{2} - 12 x + 3 \cdot 4 - 3$

Step 1.2.2.1.5.2

Multiply $3$ by $4$ .

$0 = 3 x^{2} - 12 x + 12 - 3$

Step 1.2.2.2

Subtract $3$ from $12$ .

$0 = 3 x^{2} - 12 x + 9$

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (3, 0)$

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 $y$ .

$y = 3 {((0) - 2)}^{2} - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 {(0 - 2)}^{2} - 3$

Step 2.2.2

Remove parentheses.

$y = 3 {((0) - 2)}^{2} - 3$

Step 2.2.3

Simplify $3 {((0) - 2)}^{2} - 3$ .

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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 $0$ .

$y = 3 {(- 2)}^{2} - 3$

Step 2.2.3.1.2

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

$y = 3 \cdot 4 - 3$

Step 2.2.3.1.3

Multiply $3$ by $4$ .

$y = 12 - 3$

Step 2.2.3.2

Subtract $3$ from $12$ .

$y = 9$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 9)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (3, 0)$

y-intercept(s): $(0, 9)$

Step 4