Find the Axis of Symmetry f(x)=x^2-18x+80

Solution:

Step:1

Express the given function $f(x) = x^2 - 18x + 80$ in the standard equation form as $y = x^2 - 18x + 80$.

Step:2

Transform the quadratic equation into the vertex form.

Step:2.1

Complete the square for the quadratic expression $x^2 - 18x + 80$.

Step:2.1.1

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

$$a = 1, b = -18, c = 80$$

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 of $a$ and $b$ into the equation to find $d$.

$$d = \frac{-18}{2 \cdot 1}$$

Step:2.1.3.2

Simplify the fraction by reducing common factors.

Step:2.1.3.2.1

Factor out a 2 from the numerator.

$$d = \frac{2 \cdot -9}{2 \cdot 1}$$

Step:2.1.3.2.2

Eliminate the common factors in the fraction.

Step:2.1.3.2.2.1

Factor out a 2 from the denominator.

$$d = \frac{2 \cdot -9}{2(1)}$$

Step:2.1.3.2.2.2

Remove the common factor of 2.

$$d = \frac{\cancel{2} \cdot -9}{\cancel{2} \cdot 1}$$

Step:2.1.3.2.2.3

Rewrite the simplified expression.

$$d = \frac{-9}{1}$$

Step:2.1.3.2.2.4

Divide -9 by 1 to find the value of $d$.

$$d = -9$$

Step:2.1.4

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

Step:2.1.4.1

Plug in the values for $c$, $b$, and $a$ into the equation to calculate $e$.

$$e = 80 - \frac{(-18)^2}{4 \cdot 1}$$

Step:2.1.4.2

Carry out the simplification process on the right side.

Step:2.1.4.2.1

Simplify each component separately.

Step:2.1.4.2.1.1

Square the term $-18$.

$$e = 80 - \frac{324}{4 \cdot 1}$$

Step:2.1.4.2.1.2

Multiply 4 by 1.

$$e = 80 - \frac{324}{4}$$

Step:2.1.4.2.1.3

Divide 324 by 4.

$$e = 80 - 81$$

Step:2.1.4.2.1.4

Multiply -1 by 81.

$$e = 80 - 81$$

Step:2.1.4.2.2

Subtract 81 from 80 to find the value of $e$.

$$e = -1$$

Step:2.1.5

Insert the values of $a$, $d$, and $e$ into the vertex form equation $(x - 9)^2 - 1$.

Step:2.2

Set the equation $y$ equal to the new expression on the right side.

$$y = (x - 9)^2 - 1$$

Step:3

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

$$a = 1, h = 9, k = -1$$

Step:4

Given that $a$ is positive, the parabola opens upwards.

Step:5

Locate the vertex of the parabola, which is given by the point $(h, k)$.

$$(9, -1)$$

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.

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

Step:6.3

Simplify by removing common factors.

Step:6.3.1

Eliminate the common factor of 1.

$$\frac{\cancel{1}}{4 \cdot \cancel{1}}$$

Step:6.3.2

Present the simplified expression.

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

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.

$$(h, k + p)$$

Step:7.2

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

$$(9, -\frac{3}{4})$$

Step:8

Identify the axis of symmetry, which is the line that passes through both the vertex and the focus.

$$x = 9$$

Step:9

The axis of symmetry for the given quadratic function $f(x) = x^2 - 18x + 80$ is the vertical line $x = 9$.

Knowledge Notes:

The process of finding the axis of symmetry for a quadratic function involves several key concepts:

1. Standard Form of a Quadratic Function: Any quadratic function can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$.

2. Vertex Form of a Quadratic Function: The vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for easily identifying the vertex and the axis of symmetry.

3. Completing the Square: This is a method used to rewrite a quadratic function in vertex form. It involves creating a perfect square trinomial from the quadratic expression.

4. Axis of Symmetry: For a parabola represented by a quadratic function, the axis of symmetry is a vertical line that passes through the vertex of the parabola. It divides the parabola into two mirror-image halves. The equation of the axis of symmetry can be found using the formula $x = h$, where $h$ is the x-coordinate of the vertex.

5. Vertex of a Parabola: The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. The coordinates of the vertex can be found using the vertex form of the quadratic function or by using the formulas $h = -\frac{b}{2a}$ and $k = f(h)$.

6. Focus and Directrix: The focus is a point inside the parabola, and the directrix is a line outside the parabola such that the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. The distance $p$ from the vertex to the focus (or directrix) is given by $\frac{1}{4a}$ for the parabola $y = a(x - h)^2 + k$.

Understanding these concepts is essential for analyzing and graphing quadratic functions and their properties.