Graph f(x)=4(x-2)^2+3

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

Step 1

Find the properties of the given parabola.

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

$a = 4$

$h = 2$

$k = 3$

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

Opens Up

Find the vertex $(h, k)$ .

$(2, 3)$

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.

$(2, \frac{49}{16})$

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.

$y = \frac{47}{16}$

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

Direction: Opens Up

Vertex: $(2, 3)$

Focus: $(2, \frac{49}{16})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{47}{16}$

Direction: Opens Up

Vertex: $(2, 3)$

Focus: $(2, \frac{49}{16})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{47}{16}$

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.

$f (1) = 4 {(1)}^{2} - 16 \cdot 1 + 19$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 4 \cdot 1 - 16 \cdot 1 + 19$

Multiply $4$ by $1$ .

$f (1) = 4 - 16 \cdot 1 + 19$

Multiply $- 16$ by $1$ .

$f (1) = 4 - 16 + 19$

Simplify by adding and subtracting.

Subtract $16$ from $4$ .

$f (1) = - 12 + 19$

Add $- 12$ and $19$ .

$f (1) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = 1$ is $7$ .

$y = 7$

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

$f (0) = 4 {(0)}^{2} - 16 \cdot 0 + 19$

Simplify the result.

Simplify each term.

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

$f (0) = 4 \cdot 0 - 16 \cdot 0 + 19$

Multiply $4$ by $0$ .

$f (0) = 0 - 16 \cdot 0 + 19$

Multiply $- 16$ by $0$ .

$f (0) = 0 + 0 + 19$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 19$

Add $0$ and $19$ .

$f (0) = 19$

The final answer is $19$ .

$19$

The $y$ value at $x = 0$ is $19$ .

$y = 19$

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

$f (3) = 4 {(3)}^{2} - 16 \cdot 3 + 19$

Simplify the result.

Simplify each term.

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

$f (3) = 4 \cdot 9 - 16 \cdot 3 + 19$

Multiply $4$ by $9$ .

$f (3) = 36 - 16 \cdot 3 + 19$

Multiply $- 16$ by $3$ .

$f (3) = 36 - 48 + 19$

Simplify by adding and subtracting.

Subtract $48$ from $36$ .

$f (3) = - 12 + 19$

Add $- 12$ and $19$ .

$f (3) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = 3$ is $7$ .

$y = 7$

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

$f (4) = 4 {(4)}^{2} - 16 \cdot 4 + 19$

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 $1$ .

$f (4) = 4 {(4)}^{2} - 16 \cdot 4 + 19$

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

$f (4) = 4^{1 + 2} - 16 \cdot 4 + 19$

Add $1$ and $2$ .

$f (4) = 4^{3} - 16 \cdot 4 + 19$

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

$f (4) = 64 - 16 \cdot 4 + 19$

Multiply $- 16$ by $4$ .

$f (4) = 64 - 64 + 19$

Simplify by adding and subtracting.

Subtract $64$ from $64$ .

$f (4) = 0 + 19$

Add $0$ and $19$ .

$f (4) = 19$

The final answer is $19$ .

$19$

The $y$ value at $x = 4$ is $19$ .

$y = 19$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 19 \\ 1 & 7 \\ 2 & 3 \\ 3 & 7 \\ 4 & 19 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(2, 3)$

Focus: $(2, \frac{49}{16})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{47}{16}$

$\begin{array}{c} x & y \\ 0 & 19 \\ 1 & 7 \\ 2 & 3 \\ 3 & 7 \\ 4 & 19 \end{array}$

Step 4

Graph f(x)=4(x-2)^2+3

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