Find the Axis of Symmetry f(x)=(x-5)^2-9

Solution:

Step:1

Represent the function $f(x) = (x-5)^2-9$ in standard equation form as $y = (x-5)^2-9$.

Step:2

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

• $a = 1$
• $h = 5$
• $k = -9$

Step:3

Determine the direction in which the parabola opens based on the sign of $a$.

• Since $a$ is positive, the parabola opens upwards.

Step:4

Locate the vertex of the parabola using the coordinates $(h, k)$.

• Vertex: $(5, -9)$

Step:5

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

Step:5.1

Use the formula for $p$ which is $\frac{1}{4a}$.

Step:5.2

Insert the value of $a$ into the formula to get $p$.

• $p = \frac{1}{4 \cdot 1}$

Step:5.3

Simplify the expression by removing common factors.

Step:5.3.1

Eliminate the common factor of $1$.

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

Step:5.3.2

Express the simplified form of $p$.

• $p = \frac{1}{4}$

Step:6

Determine the focus of the parabola.

Step:6.1

For a parabola that opens up or down, the focus is found by adding $p$ to the y-coordinate $k$ of the vertex.

• Focus: $(h, k + p)$

Step:6.2

Substitute the known values for $h$, $k$, and $p$ to find the focus.

• Focus: $(5, -9 + \frac{1}{4})$
• Focus: $(5, -\frac{35}{4})$

Step:7

Identify the axis of symmetry, which is the vertical line passing through both the vertex and the focus.

• Axis of Symmetry: $x = 5$

Knowledge Notes:

The problem involves finding the axis of symmetry for a given quadratic function. The quadratic function is presented in the form $f(x) = (x-5)^2-9$, which is a transformation of the standard form $y = ax^2 + bx + c$. The steps taken to solve the problem are based on the properties of parabolas and their symmetrical nature.

1. Vertex Form of a Parabola: The vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola, and $a$ determines the direction and width of the parabola. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.

2. Axis of Symmetry: For a parabola, the axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is always $x = h$, where $h$ is the x-coordinate of the vertex.

3. Focus of a Parabola: The focus is a point inside the parabola where all the reflected rays (assuming the parabola is a mirror) meet. The distance $p$ from the vertex to the focus can be calculated using the formula $p = \frac{1}{4a}$.

4. Simplifying Expressions: In the process of solving, simplification of fractions and expressions is often required to get to the final answer.

5. Parabola Opening Direction: The sign of the coefficient $a$ in the vertex form of a parabola determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$).

By following the steps outlined in the solution, one can find the axis of symmetry for any quadratic function presented in vertex form.