Graph y=3x^2-12x+10

$y = 3 x^{2} - 12 x + 10$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 12 x + 10$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 3$

$b = - 12$

$c = 10$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 12}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot - 6}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{- 6}{3}$

Cancel the common factor of $- 6$ and $3$ .

Factor $3$ out of $- 6$ .

$d = \frac{3 \cdot - 2}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$d = \frac{3 \cdot - 2}{3 (1)}$

Cancel the common factor.

$d = \frac{3 \cdot - 2}{3 \cdot 1}$

Rewrite the expression.

$d = \frac{- 2}{1}$

Divide $- 2$ by $1$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 10 - \frac{{(- 12)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = 10 - \frac{144}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 10 - \frac{144}{12}$

Divide $144$ by $12$ .

$e = 10 - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$e = 10 - 12$

Subtract $12$ from $10$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - 2)}^{2} - 2$ .

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

Set $y$ equal to the new right side.

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

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 3$

$h = 2$

$k = - 2$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(2, - 2)$

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$\frac{1}{12}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(2, - \frac{23}{12})$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 2$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{25}{12}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(2, - 2)$

Focus: $(2, - \frac{23}{12})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{25}{12}$

Direction: Opens Up

Vertex: $(2, - 2)$

Focus: $(2, - \frac{23}{12})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{25}{12}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $1$ in the expression.

$f (1) = 3 {(1)}^{2} - 12 \cdot 1 + 10$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 3 \cdot 1 - 12 \cdot 1 + 10$

Multiply $3$ by $1$ .

$f (1) = 3 - 12 \cdot 1 + 10$

Multiply $- 12$ by $1$ .

$f (1) = 3 - 12 + 10$

Simplify by adding and subtracting.

Subtract $12$ from $3$ .

$f (1) = - 9 + 10$

Add $- 9$ and $10$ .

$f (1) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 1$ is $1$ .

$y = 1$

Replace the variable $x$ with $0$ in the expression.

$f (0) = 3 {(0)}^{2} - 12 \cdot 0 + 10$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = 3 \cdot 0 - 12 \cdot 0 + 10$

Multiply $3$ by $0$ .

$f (0) = 0 - 12 \cdot 0 + 10$

Multiply $- 12$ by $0$ .

$f (0) = 0 + 0 + 10$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 10$

Add $0$ and $10$ .

$f (0) = 10$

The final answer is $10$ .

$10$

The $y$ value at $x = 0$ is $10$ .

$y = 10$

Replace the variable $x$ with $3$ in the expression.

$f (3) = 3 {(3)}^{2} - 12 \cdot 3 + 10$

Simplify the result.

Simplify each term.

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Multiply $3$ by ${(3)}^{2}$ .

Raise $3$ to the power of $1$ .

$f (3) = 3 {(3)}^{2} - 12 \cdot 3 + 10$

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

$f (3) = 3^{1 + 2} - 12 \cdot 3 + 10$

Add $1$ and $2$ .

$f (3) = 3^{3} - 12 \cdot 3 + 10$

Raise $3$ to the power of $3$ .

$f (3) = 27 - 12 \cdot 3 + 10$

Multiply $- 12$ by $3$ .

$f (3) = 27 - 36 + 10$

Simplify by adding and subtracting.

Subtract $36$ from $27$ .

$f (3) = - 9 + 10$

Add $- 9$ and $10$ .

$f (3) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 3$ is $1$ .

$y = 1$

Replace the variable $x$ with $4$ in the expression.

$f (4) = 3 {(4)}^{2} - 12 \cdot 4 + 10$

Simplify the result.

Simplify each term.

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

$f (4) = 3 \cdot 16 - 12 \cdot 4 + 10$

Multiply $3$ by $16$ .

$f (4) = 48 - 12 \cdot 4 + 10$

Multiply $- 12$ by $4$ .

$f (4) = 48 - 48 + 10$

Simplify by adding and subtracting.

Subtract $48$ from $48$ .

$f (4) = 0 + 10$

Add $0$ and $10$ .

$f (4) = 10$

The final answer is $10$ .

$10$

The $y$ value at $x = 4$ is $10$ .

$y = 10$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 10 \\ 1 & 1 \\ 2 & - 2 \\ 3 & 1 \\ 4 & 10 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(2, - 2)$

Focus: $(2, - \frac{23}{12})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{25}{12}$

$\begin{array}{c} x & y \\ 0 & 10 \\ 1 & 1 \\ 2 & - 2 \\ 3 & 1 \\ 4 & 10 \end{array}$

Step 4

Graph y=3x^2-12x+10

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