Find the Axis of Symmetry f(x)=x^2+0x-25

Solution:

Step 1:

Express the function $f(x) = x^2 + 0x - 25$ as $y = x^2 + 0x - 25$.

Step 2:

Transform the equation into the vertex form.

Step 2.1:

Simplify the equation $x^2 + 0x - 25$.

Step 2.1.1:

Eliminate the term $0x$ from the equation: $y = x^2 - 25$.

Step 2.2:

Complete the square for the expression $x^2 - 25$.

Step 2.2.1:

Identify the coefficients $a$, $b$, and $c$ from the standard form $ax^2 + bx + c$: $a=1$, $b=0$, $c=-25$.

Step 2.2.2:

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

Step 2.2.3:

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

Step 2.2.3.1:

Insert the known values for $a$ and $b$ into the formula: $d = \frac{0}{2 \cdot 1}$.

Step 2.2.3.2:

Simplify the fraction by removing the zero factor: $d = 0$.

Step 2.2.4:

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

Step 2.2.4.1:

Place the values of $c$, $b$, and $a$ into the formula: $e = -25 - \frac{0^2}{4 \cdot 1}$.

Step 2.2.4.2:

Simplify the expression to find $e$: $e = -25$.

Step 2.2.5:

Insert the values of $a$, $d$, and $e$ into the vertex form: $y = (x + 0)^2 - 25$.

Step 3:

From the vertex form $y = a(x - h)^2 + k$, identify $a=1$, $h=0$, and $k=-25$.

Step 4:

Since $a$ is positive, the parabola opens upward.

Step 5:

The vertex of the parabola is at $(h, k) = (0, -25)$.

Step 6:

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

Step 6.1:

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

Step 6.2:

Substitute the value of $a$ into the formula: $p = \frac{1}{4 \cdot 1}$.

Step 6.3:

Simplify to find $p$: $p = \frac{1}{4}$.

Step 7:

Determine the focus of the parabola.

Step 7.1:

For a parabola opening upward, add $p$ to the y-coordinate $k$ of the vertex to find the focus: $(h, k + p)$.

Step 7.2:

Insert the known values for $h$, $p$, and $k$: Focus is at $(0, -25 + \frac{1}{4})$.

Step 8:

The axis of symmetry is the line that passes through the vertex and the focus: $x = h = 0$.

Knowledge Notes:

1. Quadratic Functions: A quadratic function is typically written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola.

2. Vertex Form: The vertex form of a quadratic function 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 direction in which the parabola opens.

3. Axis of Symmetry: The axis of symmetry of a parabola is a vertical line that passes through the vertex. For a quadratic function $f(x) = ax^2 + bx + c$, the axis of symmetry is $x = -\frac{b}{2a}$.

4. Completing the Square: This is a technique used to rewrite a quadratic function in vertex form. It involves creating a perfect square trinomial from the quadratic term and the linear term, and then adjusting the constant term to maintain equality.

5. Parabola Direction: The direction in which a parabola opens is determined by the sign of the coefficient $a$. If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

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 related to $a$ by the formula $p = \frac{1}{4a}$.

7. Vertex: The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upward or downward. It is a point of symmetry for the parabola.

By understanding these concepts, we can analyze and graph quadratic functions, determine their key features, and solve related problems.