Graph y=3x^2-7x+1

$y = 3 x^{2} - 7 x + 1$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 7 x + 1$ .

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

$a = 3$

$b = - 7$

$c = 1$

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

$d = \frac{- 7}{2 \cdot 3}$

Simplify the right side.

Multiply $2$ by $3$ .

$d = \frac{- 7}{6}$

Move the negative in front of the fraction.

$d = - \frac{7}{6}$

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

$e = 1 - \frac{{(- 7)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$e = 1 - \frac{49}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 1 - \frac{49}{12}$

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

$e = \frac{12}{12} - \frac{49}{12}$

Combine the numerators over the common denominator.

$e = \frac{12 - 49}{12}$

Subtract $49$ from $12$ .

$e = \frac{- 37}{12}$

Move the negative in front of the fraction.

$e = - \frac{37}{12}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{7}{6})}^{2} - \frac{37}{12}$ .

$3 {(x - \frac{7}{6})}^{2} - \frac{37}{12}$

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{7}{6})}^{2} - \frac{37}{12}$

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

$a = 3$

$h = \frac{7}{6}$

$k = - \frac{37}{12}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{7}{6}, - \frac{37}{12})$

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.

$(\frac{7}{6}, - 3)$

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

$x = \frac{7}{6}$

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.

$y = - \frac{19}{6}$

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

Direction: Opens Up

Vertex: $(\frac{7}{6}, - \frac{37}{12})$

Focus: $(\frac{7}{6}, - 3)$

Axis of Symmetry: $x = \frac{7}{6}$

Directrix: $y = - \frac{19}{6}$

Direction: Opens Up

Vertex: $(\frac{7}{6}, - \frac{37}{12})$

Focus: $(\frac{7}{6}, - 3)$

Axis of Symmetry: $x = \frac{7}{6}$

Directrix: $y = - \frac{19}{6}$

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 $0$ in the expression.

$f (0) = 3 {(0)}^{2} - 7 \cdot 0 + 1$

Simplify the result.

Simplify each term.

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

$f (0) = 3 \cdot 0 - 7 \cdot 0 + 1$

Multiply $3$ by $0$ .

$f (0) = 0 - 7 \cdot 0 + 1$

Multiply $- 7$ by $0$ .

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

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 1$

Add $0$ and $1$ .

$f (0) = 1$

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.

$f (- 1) = 3 {(- 1)}^{2} - 7 \cdot - 1 + 1$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$f (- 1) = 3 \cdot 1 - 7 \cdot - 1 + 1$

Multiply $3$ by $1$ .

$f (- 1) = 3 - 7 \cdot - 1 + 1$

Multiply $- 7$ by $- 1$ .

$f (- 1) = 3 + 7 + 1$

Simplify by adding numbers.

Add $3$ and $7$ .

$f (- 1) = 10 + 1$

Add $10$ and $1$ .

$f (- 1) = 11$

The final answer is $11$ .

$11$

The $y$ value at $x = - 1$ is $11$ .

$y = 11$

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

$f (2) = 3 {(2)}^{2} - 7 \cdot 2 + 1$

Simplify the result.

Simplify each term.

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

$f (2) = 3 \cdot 4 - 7 \cdot 2 + 1$

Multiply $3$ by $4$ .

$f (2) = 12 - 7 \cdot 2 + 1$

Multiply $- 7$ by $2$ .

$f (2) = 12 - 14 + 1$

Simplify by adding and subtracting.

Subtract $14$ from $12$ .

$f (2) = - 2 + 1$

Add $- 2$ and $1$ .

$f (2) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = 2$ is $- 1$ .

$y = - 1$

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

$f (3) = 3 {(3)}^{2} - 7 \cdot 3 + 1$

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 $1$ .

$f (3) = 3 {(3)}^{2} - 7 \cdot 3 + 1$

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

$f (3) = 3^{1 + 2} - 7 \cdot 3 + 1$

Add $1$ and $2$ .

$f (3) = 3^{3} - 7 \cdot 3 + 1$

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

$f (3) = 27 - 7 \cdot 3 + 1$

Multiply $- 7$ by $3$ .

$f (3) = 27 - 21 + 1$

Simplify by adding and subtracting.

Subtract $21$ from $27$ .

$f (3) = 6 + 1$

Add $6$ and $1$ .

$f (3) = 7$

The final answer is $7$ .

$7$

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

$y = 7$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 11 \\ 0 & 1 \\ \frac{7}{6} & - \frac{37}{12} \\ 2 & - 1 \\ 3 & 7 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{7}{6}, - \frac{37}{12})$

Focus: $(\frac{7}{6}, - 3)$

Axis of Symmetry: $x = \frac{7}{6}$

Directrix: $y = - \frac{19}{6}$

$\begin{array}{c} x & y \\ - 1 & 11 \\ 0 & 1 \\ \frac{7}{6} & - \frac{37}{12} \\ 2 & - 1 \\ 3 & 7 \end{array}$

Step 4

Graph y=3x^2-7x+1

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