Find the Axis of Symmetry (x+4)^2-2
The question is asking for the axis of symmetry for a quadratic function represented by the equation (x+4)^2-2. The axis of symmetry is a vertical line that divides the parabola into two mirror images, each reflecting the other. It pertains to the value of 'x' where this line is located on the Cartesian coordinate system. The equation given is in vertex form, which makes it easier to identify the vertex of the parabola, from which the axis of symmetry can be derived. The axis of symmetry can be found by setting the value inside the parentheses equal to zero, as this represents the 'x' value of the vertex.
$\left(\left(\right. x + 4 \left.\right)\right)^{2} - 2$
Answer
Solution:
Step 1:
Express the given function $y = (x + 4)^2 - 2$.
Step 2:
Identify the coefficients in the standard vertex form $y = a(x - h)^2 + k$ where $a = 1$, $h = -4$, and $k = -2$.
Step 3:
Determine the orientation of the parabola. Since $a > 0$, the parabola opens upwards.
Step 4:
The vertex of the parabola is at the point $(h, k)$, which is $(-4, -2)$.
Step 5:
Calculate the distance $p$ from the vertex to the focus of the parabola.
Step 5.1:
Use the formula $p = \frac{1}{4a}$ to find $p$.
Step 5.2:
Plug in the value of $a$ to get $p = \frac{1}{4 \cdot 1}$.
Step 5.3:
Simplify the expression by removing common factors.
Step 5.3.1:
Eliminate the common factor to obtain $p = \frac{1}{4}$.
Step 5.3.2:
Finalize the expression for $p$ as $p = \frac{1}{4}$.
Step 6:
Locate the focus of the parabola.
Step 6.1:
To find the focus, add $p$ to the y-coordinate of the vertex for a parabola that opens up or down, yielding $(h, k + p)$.
Step 6.2:
Substitute the known values for $h$, $p$, and $k$ to find the focus at $(-4, -2 + \frac{1}{4})$, simplifying to $(-4, -\frac{7}{4})$.
Step 7:
The axis of symmetry is the vertical line that passes through both the vertex and the focus, which is $x = -4$.
Knowledge Notes:
The problem involves finding the axis of symmetry for a quadratic function in 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, and $a$ determines the direction of the opening (upwards if $a > 0$, downwards if $a < 0$).
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a parabola in vertex form, the axis of symmetry is always the line $x = h$, where $h$ is the x-coordinate of the vertex.
The focus of a parabola is a point located inside the parabola, a distance $p$ from the vertex along the axis of symmetry. For a parabola that opens upwards or downwards, the focus is at $(h, k + p)$ or $(h, k - p)$, respectively. The value of $p$ can be calculated using the formula $p = \frac{1}{4a}$.
In this problem, the given quadratic function is already in vertex form, making it straightforward to identify the vertex and the axis of symmetry. The focus is then found using the value of $a$, and the axis of symmetry is confirmed as the line passing through both the vertex and the focus.
