Graph y=4x^2+5x-1

$y = 4 x^{2} + 5 x - 1$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} + 5 x - 1$ .

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

$a = 4$

$b = 5$

$c = - 1$

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

$d = \frac{5}{2 \cdot 4}$

Multiply $2$ by $4$ .

$d = \frac{5}{8}$

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

$e = - 1 - \frac{5^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $5$ to the power of $2$ .

$e = - 1 - \frac{25}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = - 1 - \frac{25}{16}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$e = - 1 \cdot \frac{16}{16} - \frac{25}{16}$

Combine $- 1$ and $\frac{16}{16}$ .

$e = \frac{- 1 \cdot 16}{16} - \frac{25}{16}$

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 16 - 25}{16}$

Simplify the numerator.

Multiply $- 1$ by $16$ .

$e = \frac{- 16 - 25}{16}$

Subtract $25$ from $- 16$ .

$e = \frac{- 41}{16}$

Move the negative in front of the fraction.

$e = - \frac{41}{16}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$ .

$4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$

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

$a = 4$

$h = - \frac{5}{8}$

$k = - \frac{41}{16}$

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

Opens Up

Find the vertex $(h, k)$ .

$(- \frac{5}{8}, - \frac{41}{16})$

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.

$(- \frac{5}{8}, - \frac{5}{2})$

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

$x = - \frac{5}{8}$

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.

$y = - \frac{21}{8}$

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

Direction: Opens Up

Vertex: $(- \frac{5}{8}, - \frac{41}{16})$

Focus: $(- \frac{5}{8}, - \frac{5}{2})$

Axis of Symmetry: $x = - \frac{5}{8}$

Directrix: $y = - \frac{21}{8}$

Direction: Opens Up

Vertex: $(- \frac{5}{8}, - \frac{41}{16})$

Focus: $(- \frac{5}{8}, - \frac{5}{2})$

Axis of Symmetry: $x = - \frac{5}{8}$

Directrix: $y = - \frac{21}{8}$

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 $- 2$ in the expression.

$f (- 2) = 4 {(- 2)}^{2} + 5 (- 2) - 1$

Simplify the result.

Simplify each term.

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

$f (- 2) = 4 \cdot 4 + 5 (- 2) - 1$

Multiply $4$ by $4$ .

$f (- 2) = 16 + 5 (- 2) - 1$

Multiply $5$ by $- 2$ .

$f (- 2) = 16 - 10 - 1$

Simplify by subtracting numbers.

Subtract $10$ from $16$ .

$f (- 2) = 6 - 1$

Subtract $1$ from $6$ .

$f (- 2) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = - 2$ is $5$ .

$y = 5$

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

$f (- 3) = 4 {(- 3)}^{2} + 5 (- 3) - 1$

Simplify the result.

Simplify each term.

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

$f (- 3) = 4 \cdot 9 + 5 (- 3) - 1$

Multiply $4$ by $9$ .

$f (- 3) = 36 + 5 (- 3) - 1$

Multiply $5$ by $- 3$ .

$f (- 3) = 36 - 15 - 1$

Simplify by subtracting numbers.

Subtract $15$ from $36$ .

$f (- 3) = 21 - 1$

Subtract $1$ from $21$ .

$f (- 3) = 20$

The final answer is $20$ .

$20$

The $y$ value at $x = - 3$ is $20$ .

$y = 20$

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

$f (0) = 4 {(0)}^{2} + 5 (0) - 1$

Simplify the result.

Simplify each term.

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

$f (0) = 4 \cdot 0 + 5 (0) - 1$

Multiply $4$ by $0$ .

$f (0) = 0 + 5 (0) - 1$

Multiply $5$ by $0$ .

$f (0) = 0 + 0 - 1$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 1$

Subtract $1$ from $0$ .

$f (0) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = 0$ is $- 1$ .

$y = - 1$

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

$f (1) = 4 {(1)}^{2} + 5 (1) - 1$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 4 \cdot 1 + 5 (1) - 1$

Multiply $4$ by $1$ .

$f (1) = 4 + 5 (1) - 1$

Multiply $5$ by $1$ .

$f (1) = 4 + 5 - 1$

Simplify by adding and subtracting.

Add $4$ and $5$ .

$f (1) = 9 - 1$

Subtract $1$ from $9$ .

$f (1) = 8$

The final answer is $8$ .

$8$

The $y$ value at $x = 1$ is $8$ .

$y = 8$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 20 \\ - 2 & 5 \\ - \frac{5}{8} & - \frac{41}{16} \\ 0 & - 1 \\ 1 & 8 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{5}{8}, - \frac{41}{16})$

Focus: $(- \frac{5}{8}, - \frac{5}{2})$

Axis of Symmetry: $x = - \frac{5}{8}$

Directrix: $y = - \frac{21}{8}$

$\begin{array}{c} x & y \\ - 3 & 20 \\ - 2 & 5 \\ - \frac{5}{8} & - \frac{41}{16} \\ 0 & - 1 \\ 1 & 8 \end{array}$

Step 4

Graph y=4x^2+5x-1

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