To find the equation of a parabola given its axis of symmetry and two points that it passes through, we can start by recognizing that for a parabola with its axis of symmetry at \( x = h \), the equation can be written in the form:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
Given that the axis of symmetry is \( x = -3 \), we know that the vertex lies on this line. Therefore, the x-coordinate of the vertex is -3.
We also have two points that the parabola passes through: (-5, 0) and (-3, -4).
Let's start by finding the vertex using the axis of symmetry. Since the vertex lies on the line \( x = -3 \), the x-coordinate of the vertex is -3.
Now, let's find the y-coordinate of the vertex. Since the parabola is symmetrical, the y-coordinate of the vertex will be the midpoint of the y-coordinates of the given points. So,
\[ y_{\text{vertex}} = \frac{0 + (-4)}{2} = -2 \]
So, the vertex of the parabola is (-3, -2).
Now, let's substitute the vertex coordinates into the equation:
\[ y = a(x - (-3))^2 - 2 \]
\[ y = a(x + 3)^2 - 2 \]
Now, we need to find the value of \( a \). We can use one of the given points to solve for \( a \). Let's use the point (-5, 0):
\[ 0 = a(-5 + 3)^2 - 2 \]
\[ 0 = a(-2)^2 - 2 \]
\[ 0 = 4a - 2 \]
\[ 4a = 2 \]
\[ a = \frac{1}{2} \]
Now we have found the value of \( a \), which is \( \frac{1}{2} \).
Therefore, the equation of the parabola is:
\[ y = \frac{1}{2}(x + 3)^2 - 2 \]
