Find the Axis of Symmetry f(x)=2x^2+10x+12

Solution:

Step:1

Represent the quadratic function $f(x) = 2x^2 + 10x + 12$ with $y$.

$$y = 2x^2 + 10x + 12$$

Step:2

Transform the quadratic equation into the vertex form.

Step:2.1

Complete the square for the quadratic expression $2x^2 + 10x + 12$.

Step:2.1.1

Identify the coefficients $a$, $b$, and $c$ from the standard form $ax^2 + bx + c$.

$$a = 2, b = 10, c = 12$$

Step:2.1.2

Consider the standard vertex form of a quadratic $a(x + d)^2 + e$.

Step:2.1.3

Calculate $d$ using $d = -\frac{b}{2a}$.

Step:2.1.3.1

Insert $a$ and $b$ into the formula to find $d$.

$$d = -\frac{10}{2 \cdot 2}$$

Step:2.1.3.2

Simplify the fraction by reducing common factors.

Step:2.1.3.2.1

Extract the factor of $2$ from the numerator.

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

Step:2.1.3.2.2

Eliminate the common factors in the fraction.

Step:2.1.3.2.2.1

Isolate the factor of $2$ in the denominator.

$$d = -\frac{2 \cdot 5}{2(2)}$$

Step:2.1.3.2.2.2

Remove the common factor of $2$.

$$d = -\frac{\cancel{2} \cdot 5}{\cancel{2} \cdot 2}$$

Step:2.1.3.2.2.3

Rewrite the simplified expression for $d$.

$$d = -\frac{5}{2}$$

Step:2.1.4

Determine $e$ using the formula $e = c - a(d)^2$.

Step:2.1.4.1

Plug in the values for $a$, $b$, and $c$ to find $e$.

$$e = 12 - 2\left(-\frac{5}{2}\right)^2$$

Step:2.1.4.2

Simplify the right-hand side of the equation.

Step:2.1.4.2.1

Perform the operations in each term.

Step:2.1.4.2.1.1

Square $-\frac{5}{2}$.

$$e = 12 - 2\left(\frac{25}{4}\right)$$

Step:2.1.4.2.1.2

Multiply $2$ by $\frac{25}{4}$.

$$e = 12 - \frac{50}{4}$$

Step:2.1.4.2.1.3

Reduce the fraction by eliminating common factors.

Step:2.1.4.2.1.3.1

Factor out $2$ from the numerator.

$$e = 12 - \frac{2(25)}{4}$$

Step:2.1.4.2.1.3.2

Remove the common factors.

Step:2.1.4.2.1.3.2.1

Isolate the factor of $2$ in the denominator.

$$e = 12 - \frac{2 \cdot 25}{2 \cdot 2}$$

Step:2.1.4.2.1.3.2.2

Cancel out the common factor of $2$.

$$e = 12 - \frac{\cancel{2} \cdot 25}{\cancel{2} \cdot 2}$$

Step:2.1.4.2.1.3.2.3

Express the simplified form of $e$.

$$e = 12 - \frac{25}{2}$$

Step:2.1.4.2.2

Convert $12$ to a fraction with the same denominator as $\frac{25}{2}$.

$$e = \frac{12 \cdot 2}{2} - \frac{25}{2}$$

Step:2.1.4.2.3

Combine the fractions over a common denominator.

$$e = \frac{24 - 25}{2}$$

Step:2.1.4.2.4

Simplify the numerator.

$$e = \frac{-1}{2}$$

Step:2.1.5

Insert the values of $a$, $d$, and $e$ into the vertex form equation.

$$y = 2\left(x - \frac{5}{2}\right)^2 - \frac{1}{2}$$

Step:2.2

Set $y$ equal to the transformed equation.

$$y = 2\left(x - \frac{5}{2}\right)^2 - \frac{1}{2}$$

Step:3

Identify $a$, $h$, and $k$ from the vertex form $y = a(x - h)^2 + k$.

$$a = 2, h = \frac{5}{2}, k = -\frac{1}{2}$$

Step:4

The parabola opens upwards since $a$ is positive.

Step:5

Determine the vertex $(h, k)$.

$$(\frac{5}{2}, -\frac{1}{2})$$

Step:6

Calculate $p$, the distance from the vertex to the focus of the parabola.

Step:6.1

Use the formula for the distance from the vertex to the focus $p = \frac{1}{4a}$.

Step:6.2

Substitute the value of $a$ into the formula.

$$p = \frac{1}{4 \cdot 2}$$

Step:6.3

Simplify the expression.

$$p = \frac{1}{8}$$

Step:7

Locate the focus of the parabola.

Step:7.1

The focus is found by adding $p$ to the $k$ coordinate of the vertex for a parabola that opens up.

$$(h, k + p)$$

Step:7.2

Insert the known values for $h$, $k$, and $p$ and simplify.

$$(\frac{5}{2}, -\frac{1}{2} + \frac{1}{8})$$ $$(\frac{5}{2}, -\frac{3}{8})$$

Step:8

The axis of symmetry is the line that passes through the vertex and the focus.

$$x = \frac{5}{2}$$

Step:9

The axis of symmetry for the given quadratic function is $x = \frac{5}{2}$.

Knowledge Notes:

To find the axis of symmetry for a quadratic function in the form $f(x) = ax^2 + bx + c$, you can use the formula $x = -\frac{b}{2a}$. This formula derives from the process of completing the square, which involves rewriting the quadratic in vertex form, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides it into two mirror-image halves. For a parabola that opens upwards (when $a > 0$) or downwards (when $a < 0$), the axis of symmetry will always be the line $x = h$.