https://file.aiquickdraw.com/mathbot/file/17262074504982eww9xbj.jpg

The image contains a graph and several questions based on a quadratic function \( z(x) \). Let's analyze and solve the problems listed.

### From the graph:
- The quadratic curve opens upwards, indicating that the coefficient of \( x^2 \) is positive.
- The vertex is the lowest point on the curve, and from the graph, it appears to be at \( (10, -50) \).
- The vertical intercept (where the graph crosses the y-axis) appears to be at \( z(0) = 50 \).

### Now let's solve the questions one by one:

**(a) What is the vertex of the graph of \( z \)?**
- From the graph, the vertex is at \( (10, -50) \).

**(b) Write an equation for \( z \).**
- Since the vertex form of a parabola is \( z(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex, we have:
  \[
  z(x) = a(x - 10)^2 - 50
  \]
  To find \( a \), use another point from the graph, such as \( z(0) = 50 \):
  \[
  50 = a(0 - 10)^2 - 50
  \]
  \[
  50 = 100a - 50
  \]
  \[
  100 = 100a
  \]
  \[
  a = 1
  \]
  Therefore, the equation is \( z(x) = (x - 10)^2 - 50 \).

**(c) Solve \( z(x) = 50 \).**
- Set \( z(x) = 50 \) and solve:
  \[
  (x - 10)^2 - 50 = 50
  \]
  \[
  (x - 10)^2 = 100
  \]
  Taking the square root of both sides:
  \[
  x - 10 = \pm 10
  \]
  \[
  x = 10 \pm 10
  \]
  Thus, \( x = 20 \) or \( x = 0 \). Therefore, \( z(x) = 50 \) at \( x = 0 \) and \( x = 20 \).

**(d) Solve \( z(x) \geq 50 \).**
- We already know \( z(x) = 50 \) at \( x = 0 \) and \( x = 20 \). Since the parabola opens upwards, \( z(x) \geq 50 \) outside this interval:
  \[
  x \leq 0 \quad \text{or} \quad x \geq 20
  \]

**(e) Find the \( x \)-intercepts of \( z \).**
- The \( x \)-intercepts occur when \( z(x) = 0 \):
  \[
  (x - 10)^2 - 50 = 0
  \]
  \[
  (x - 10)^2 = 50
  \]
  Taking the square root:
  \[
  x - 10 = \pm \sqrt{50}
  \]
  \[
  x = 10 \pm \sqrt{50}
  \]
  Approximate \( \sqrt{50} \approx 7.07 \), so the intercepts are:
  \[
  x \approx 10 + 7.07 = 17.07 \quad \text{and} \quad x \approx 10 - 7.07 = 2.93
  \]
  Thus, the \( x \)-intercepts are approximately \( x = 2.93 \) and \( x = 17.07 \).

**(f) Find the vertical intercept of \( z \).**
- The vertical intercept is the y-value when \( x = 0 \):
  \[
  z(0) = (0 - 10)^2 - 50 = 100 - 50 = 50
  \]
  So, the vertical intercept is \( (0, 50) \).

**(g) What are the domain and range of \( z \)?**
- **Domain:** The parabola continues indefinitely left and right, so the domain is all real numbers, \( (-\infty, \infty) \).
- **Range:** The lowest point of the graph is the vertex at \( z = -50 \), and the parabola opens upwards, so the range is \( [-50, \infty) \).

---

Do you want more details or have any questions about the solutions? Here are five related questions to expand your understanding:

1. How do you convert a quadratic function from vertex form to standard form?
2. What is the relationship between the discriminant and the number of x-intercepts of a quadratic function?
3. How can you determine the direction of a parabola based on its equation?
4. What is the significance of the axis of symmetry in a quadratic graph?
5. How would the equation change if the parabola were shifted up or down?

**Tip:** Always double-check the points from the graph when solving quadratic equations; precise values will lead to more accurate solutions.

Use the graph of z(x) below to answer the following questions:
(a) What is the vertex of the graph of z?
(b) Write an equation for z.
(c) Solve z(x) = 50.
(d) Solve z(x) ≥ 50.
(e) Find the x-intercepts of z.
(f) Find the vertical intercept of z.
(g) What are the domain and range of z?

Learn how to solve quadratic function problems using graphs. This involves finding the vertex, solving inequalities, and determining the domain and range of the function z(x). Suitable for Grades 9-11.

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