Question

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Explanation:

Identify the given information

The problem asks: "The vertex of a parabola is \((-3, -4)\). One x-intercept is \(7\). What is the other x-intercept?"
We are given:

• The vertex of the parabola: \((h, k) = (-3, -4)\)
• One x-intercept: \(x_1 = 7\), which corresponds to the point \((7, 0)\)

Apply the symmetry of a parabola

A parabola is perfectly symmetric with respect to its axis of symmetry.
The axis of symmetry is a vertical line passing through the vertex:

$$x = h$$

For this parabola, the axis of symmetry is:

$$x = -3$$

Calculate the distance to the axis

Since the parabola is symmetric, any two points with the same y-coordinate (such as the two x-intercepts, where \(y = 0\)) must be at equal horizontal distances from the axis of symmetry.
Let's find the horizontal distance from the given x-intercept \(x_1 = 7\) to the axis of symmetry \(x = -3\):

$$\text{Distance} = |x_1 - h| = |7 - (-3)| = |7 + 3| = 10$$

Find the other x-intercept

The other x-intercept \(x_2\) must lie at the same horizontal distance of \(10\) units on the opposite side of the axis of symmetry.
Since \(x_1 = 7\) is to the right of the axis \(x = -3\), the other intercept \(x_2\) must be to the left:

$$x_2 = h - \text{Distance}$$

$$x_2 = -3 - 10 = -13$$

Answer:

The other x-intercept is \(-13\).