Graph f(x)=-1/4(x-1)^2+4

$f (x) = - \frac{1}{4} {(x - 1)}^{2} + 4$

Step 1

Find the properties of the given parabola.

Isolate $y$ to the left side of the equation.

Combine ${(x - 1)}^{2}$ and $\frac{1}{4}$ .

$y = - \frac{{(x - 1)}^{2}}{4} + 4$

Reorder terms.

$y = - \frac{1}{4} \cdot {(x - 1)}^{2} + 4$

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

$a = - \frac{1}{4}$

$h = 1$

$k = 4$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(1, 4)$

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.

$\frac{1}{4 \cdot (- \frac{1}{4})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{4})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{4})}$

Combine $4$ and $\frac{1}{4}$ .

$- \frac{1}{\frac{4}{4}}$

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$

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.

$(1, 3)$

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

$x = 1$

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.

$y = 5$

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

Direction: Opens Down

Vertex: $(1, 4)$

Focus: $(1, 3)$

Axis of Symmetry: $x = 1$

Directrix: $y = 5$

Direction: Opens Down

Vertex: $(1, 4)$

Focus: $(1, 3)$

Axis of Symmetry: $x = 1$

Directrix: $y = 5$

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.

$f (- 1) = - \frac{{(- 1)}^{2}}{4} + \frac{- 1}{2} + \frac{15}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 1) = \frac{- {(- 1)}^{2} + 15}{4} + \frac{- 1}{2}$

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

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

$f (- 1) = \frac{(- 1) {(- 1)}^{2} + 15}{4} + \frac{- 1}{2}$

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

$f (- 1) = \frac{{(- 1)}^{1 + 2} + 15}{4} + \frac{- 1}{2}$

Add $1$ and $2$ .

$f (- 1) = \frac{{(- 1)}^{3} + 15}{4} + \frac{- 1}{2}$

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

$f (- 1) = \frac{- 1 + 15}{4} + \frac{- 1}{2}$

Add $- 1$ and $15$ .

$f (- 1) = \frac{14}{4} + \frac{- 1}{2}$

Simplify each term.

Cancel the common factor of $14$ and $4$ .

Factor $2$ out of $14$ .

$f (- 1) = \frac{2 (7)}{4} + \frac{- 1}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 1) = \frac{2 \cdot 7}{2 \cdot 2} + \frac{- 1}{2}$

Cancel the common factor.

$f (- 1) = \frac{2 \cdot 7}{2 \cdot 2} + \frac{- 1}{2}$

Rewrite the expression.

$f (- 1) = \frac{7}{2} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (- 1) = \frac{7}{2} - \frac{1}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$f (- 1) = \frac{7 - 1}{2}$

Simplify the expression.

Subtract $1$ from $7$ .

$f (- 1) = \frac{6}{2}$

Divide $6$ by $2$ .

$f (- 1) = 3$

The final answer is $3$ .

$3$

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

$y = 3$

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

$f (0) = - \frac{{(0)}^{2}}{4} + \frac{0}{2} + \frac{15}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (0) = \frac{- {(0)}^{2} + 15}{4} + \frac{0}{2}$

Simplify each term.

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

$f (0) = \frac{- 0 + 15}{4} + \frac{0}{2}$

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 + 15}{4} + \frac{0}{2}$

Simplify the expression.

Add $0$ and $15$ .

$f (0) = \frac{15}{4} + \frac{0}{2}$

Divide $0$ by $2$ .

$f (0) = \frac{15}{4} + 0$

Add $\frac{15}{4}$ and $0$ .

$f (0) = \frac{15}{4}$

The final answer is $\frac{15}{4}$ .

$\frac{15}{4}$

The $y$ value at $x = 0$ is $\frac{15}{4}$ .

$y = \frac{15}{4}$

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

$f (3) = - \frac{{(3)}^{2}}{4} + \frac{3}{2} + \frac{15}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (3) = \frac{- {(3)}^{2} + 15}{4} + \frac{3}{2}$

Simplify each term.

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

$f (3) = \frac{- 1 \cdot 9 + 15}{4} + \frac{3}{2}$

Multiply $- 1$ by $9$ .

$f (3) = \frac{- 9 + 15}{4} + \frac{3}{2}$

Simplify terms.

Add $- 9$ and $15$ .

$f (3) = \frac{6}{4} + \frac{3}{2}$

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

Factor $2$ out of $6$ .

$f (3) = \frac{2 (3)}{4} + \frac{3}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (3) = \frac{2 \cdot 3}{2 \cdot 2} + \frac{3}{2}$

Cancel the common factor.

$f (3) = \frac{2 \cdot 3}{2 \cdot 2} + \frac{3}{2}$

Rewrite the expression.

$f (3) = \frac{3}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$f (3) = \frac{3 + 3}{2}$

Simplify the expression.

Add $3$ and $3$ .

$f (3) = \frac{6}{2}$

Divide $6$ by $2$ .

$f (3) = 3$

The final answer is $3$ .

$3$

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

$y = 3$

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{{(2)}^{2}}{4} + \frac{2}{2} + \frac{15}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (2) = \frac{- {(2)}^{2} + 15}{4} + \frac{2}{2}$

Simplify each term.

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

$f (2) = \frac{- 1 \cdot 4 + 15}{4} + \frac{2}{2}$

Multiply $- 1$ by $4$ .

$f (2) = \frac{- 4 + 15}{4} + \frac{2}{2}$

Simplify the expression.

Add $- 4$ and $15$ .

$f (2) = \frac{11}{4} + \frac{2}{2}$

Divide $2$ by $2$ .

$f (2) = \frac{11}{4} + 1$

Write $1$ as a fraction with a common denominator.

$f (2) = \frac{11}{4} + \frac{4}{4}$

Combine the numerators over the common denominator.

$f (2) = \frac{11 + 4}{4}$

Add $11$ and $4$ .

$f (2) = \frac{15}{4}$

The final answer is $\frac{15}{4}$ .

$\frac{15}{4}$

The $y$ value at $x = 2$ is $\frac{15}{4}$ .

$y = \frac{15}{4}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 3 \\ 0 & \frac{15}{4} \\ 1 & 4 \\ 2 & \frac{15}{4} \\ 3 & 3 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, 4)$

Focus: $(1, 3)$

Axis of Symmetry: $x = 1$

Directrix: $y = 5$

$\begin{array}{c} x & y \\ - 1 & 3 \\ 0 & \frac{15}{4} \\ 1 & 4 \\ 2 & \frac{15}{4} \\ 3 & 3 \end{array}$

Step 4

Graph f(x)=-1/4(x-1)^2+4

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