Graph y=4x^2-12x+9

$y = 4 x^{2} - 12 x + 9$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} - 12 x + 9$ .

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

$a = 4$

$b = - 12$

$c = 9$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 12}{2 \cdot 4}$

Simplify the right side.

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

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 6}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{- 6}{4}$

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 (- 3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 9 - \frac{{(- 12)}^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

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

$e = 9 - \frac{144}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 9 - \frac{144}{16}$

Divide $144$ by $16$ .

$e = 9 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 9 - 9$

Subtract $9$ from $9$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - \frac{3}{2})}^{2} + 0$ .

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

Set $y$ equal to the new right side.

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

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

$a = 4$

$h = \frac{3}{2}$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{3}{2}, 0)$

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.

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

Multiply $4$ by $4$ .

$\frac{1}{16}$

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.

$(\frac{3}{2}, \frac{1}{16})$

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

$x = \frac{3}{2}$

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.

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

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

Direction: Opens Up

Vertex: $(\frac{3}{2}, 0)$

Focus: $(\frac{3}{2}, \frac{1}{16})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1}{16}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, 0)$

Focus: $(\frac{3}{2}, \frac{1}{16})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1}{16}$

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.

$f (1) = 4 {(1)}^{2} - 12 \cdot 1 + 9$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 4 \cdot 1 - 12 \cdot 1 + 9$

Multiply $4$ by $1$ .

$f (1) = 4 - 12 \cdot 1 + 9$

Multiply $- 12$ by $1$ .

$f (1) = 4 - 12 + 9$

Simplify by adding and subtracting.

Subtract $12$ from $4$ .

$f (1) = - 8 + 9$

Add $- 8$ and $9$ .

$f (1) = 1$

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.

$f (0) = 4 {(0)}^{2} - 12 \cdot 0 + 9$

Simplify the result.

Simplify each term.

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

$f (0) = 4 \cdot 0 - 12 \cdot 0 + 9$

Multiply $4$ by $0$ .

$f (0) = 0 - 12 \cdot 0 + 9$

Multiply $- 12$ by $0$ .

$f (0) = 0 + 0 + 9$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 9$

Add $0$ and $9$ .

$f (0) = 9$

The final answer is $9$ .

$9$

The $y$ value at $x = 0$ is $9$ .

$y = 9$

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

$f (3) = 4 {(3)}^{2} - 12 \cdot 3 + 9$

Simplify the result.

Simplify each term.

Raise $3$ to the power of $2$ .

$f (3) = 4 \cdot 9 - 12 \cdot 3 + 9$

Multiply $4$ by $9$ .

$f (3) = 36 - 12 \cdot 3 + 9$

Multiply $- 12$ by $3$ .

$f (3) = 36 - 36 + 9$

Simplify by adding and subtracting.

Subtract $36$ from $36$ .

$f (3) = 0 + 9$

Add $0$ and $9$ .

$f (3) = 9$

The final answer is $9$ .

$9$

The $y$ value at $x = 3$ is $9$ .

$y = 9$

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

$f (4) = 4 {(4)}^{2} - 12 \cdot 4 + 9$

Simplify the result.

Simplify each term.

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

Multiply $4$ by ${(4)}^{2}$ .

Raise $4$ to the power of $1$ .

$f (4) = 4 {(4)}^{2} - 12 \cdot 4 + 9$

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

$f (4) = 4^{1 + 2} - 12 \cdot 4 + 9$

Add $1$ and $2$ .

$f (4) = 4^{3} - 12 \cdot 4 + 9$

Raise $4$ to the power of $3$ .

$f (4) = 64 - 12 \cdot 4 + 9$

Multiply $- 12$ by $4$ .

$f (4) = 64 - 48 + 9$

Simplify by adding and subtracting.

Subtract $48$ from $64$ .

$f (4) = 16 + 9$

Add $16$ and $9$ .

$f (4) = 25$

The final answer is $25$ .

$25$

The $y$ value at $x = 4$ is $25$ .

$y = 25$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 9 \\ 1 & 1 \\ \frac{3}{2} & 0 \\ 3 & 9 \\ 4 & 25 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, 0)$

Focus: $(\frac{3}{2}, \frac{1}{16})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1}{16}$

$\begin{array}{c} x & y \\ 0 & 9 \\ 1 & 1 \\ \frac{3}{2} & 0 \\ 3 & 9 \\ 4 & 25 \end{array}$

Step 4

Graph y=4x^2-12x+9

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