Find the Axis of Symmetry f(x)=x^2+8x

Solution:

Step:1

Express the function $f(x) = x^2 + 8x$ in the standard equation form: $y = x^2 + 8x$.

Step:2

Transform the quadratic equation into the vertex form.

Step:2.1

Complete the square for the expression $x^2 + 8x$.

Step:2.1.1

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

$$a = 1$$ $$b = 8$$ $$c = 0$$

Step:2.1.2

Recall the vertex form of a quadratic equation: $a(x + d)^2 + e$.

Step:2.1.3

Calculate the value of $d$ using the equation $d = \frac{b}{2a}$.

Step:2.1.3.1

Insert the known values for $a$ and $b$ into the equation $d = \frac{b}{2a}$.

$$d = \frac{8}{2 \cdot 1}$$

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.

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

Step:2.1.3.2.2

Eliminate the common factors.

Step:2.1.3.2.2.1

Factor out 2 from the denominator.

$$d = \frac{2 \cdot 4}{2(1)}$$

Step:2.1.3.2.2.2

Remove the common factor of 2.

$$d = \frac{\cancel{2} \cdot 4}{\cancel{2} \cdot 1}$$

Step:2.1.3.2.2.3

Rewrite the simplified expression.

$$d = \frac{4}{1}$$

Step:2.1.3.2.2.4

Perform the division of 4 by 1.

$$d = 4$$

Step:2.1.4

Determine the value of $e$ using the equation $e = c - \frac{b^2}{4a}$.

Step:2.1.4.1

Place the values of $c$, $b$, and $a$ into the equation $e = c - \frac{b^2}{4a}$.

$$e = 0 - \frac{8^2}{4 \cdot 1}$$

Step:2.1.4.2

Simplify the expression on the right side.

Step:2.1.4.2.1

Simplify each component.

Step:2.1.4.2.1.1

Square the number 8.

$$e = 0 - \frac{64}{4 \cdot 1}$$

Step:2.1.4.2.1.2

Multiply 4 by 1.

$$e = 0 - \frac{64}{4}$$

Step:2.1.4.2.1.3

Divide 64 by 4.

$$e = 0 - 16$$

Step:2.1.4.2.1.4

Multiply -1 by 16.

$$e = -16$$

Step:2.1.4.2.2

Subtract 16 from 0.

$$e = -16$$

Step:2.1.5

Insert the calculated values of $a$, $d$, and $e$ into the vertex form equation: $(x + 4)^2 - 16$.

Step:2.2

Set $y$ equal to the modified expression on the right side.

$$y = (x + 4)^2 - 16$$

Step:3

Utilize the vertex form $y = a(x - h)^2 + k$ to find the values of $a$, $h$, and $k$.

$$a = 1$$ $$h = -4$$ $$k = -16$$

Step:4

As the coefficient $a$ is positive, the parabola opens upwards.

Step:5

Identify the vertex $(h, k)$.

$$(h, k) = (-4, -16)$$

Step:6

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

Step:6.1

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

Step:6.2

Substitute the value of $a$ into the formula.

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

Step:6.3

Eliminate the common factor of 1.

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

Step:7

Determine the focus of the parabola.

Step:7.1

To find the focus, add $p$ to the y-coordinate $k$ of the vertex if the parabola opens upwards or downwards.

$$(h, k + p)$$

Step:7.2

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

$$(h, k + p) = (-4, -\frac{63}{4})$$

Step:8

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

$$x = -4$$

Step:9

The axis of symmetry for the given quadratic function $f(x) = x^2 + 8x$ is the vertical line $x = -4$.

Knowledge Notes:

To find the axis of symmetry for a quadratic function, one can use the vertex form of a quadratic equation, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry is a vertical line that passes through the vertex, and its equation is $x = h$.

The process of completing the square involves manipulating a quadratic equation into the vertex form. This is done by adding and subtracting a particular value inside the squared term to create a perfect square trinomial, which then simplifies to the vertex form.

The vertex form provides valuable information about the parabola, including the direction it opens (upward for $a > 0$, downward for $a < 0$), the vertex of the parabola, and the axis of symmetry.

The focus of a parabola is a fixed point used in the geometric definition of the curve and is related to the vertex by the distance $p = \frac{1}{4a}$. The directrix is a line perpendicular to the axis of symmetry and is the same distance $p$ from the vertex as the focus, but on the opposite side.

In this problem, we used algebraic manipulation to rewrite the given quadratic function in vertex form, from which we could easily determine the axis of symmetry.