For the quadratic function 𝑦 = 3𝑥2 − 6𝑥 − 24, find the following:
A. Vertex B. Horizontal intercepts
C. Range D. Vertical Intercept

Let's break down the given quadratic function \( y = 3x^2 - 6x - 24 \) and find the required parts:

### A. Vertex
The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula:

\[
x_{vertex} = \frac{-b}{2a}
\]

For the given function \( y = 3x^2 - 6x - 24 \), we have:
- \( a = 3 \)
- \( b = -6 \)

Using the vertex formula:

\[
x_{vertex} = \frac{-(-6)}{2(3)} = \frac{6}{6} = 1
\]

Now, substitute \( x = 1 \) back into the original equation to find \( y_{vertex} \):

\[
y_{vertex} = 3(1)^2 - 6(1) - 24 = 3 - 6 - 24 = -27
\]

Thus, the vertex is \( (1, -27) \).

---

### B. Horizontal Intercepts (Roots)
To find the horizontal intercepts (where \( y = 0 \)), solve the quadratic equation:

\[
3x^2 - 6x - 24 = 0
\]

First, divide the equation by 3 to simplify:

\[
x^2 - 2x - 8 = 0
\]

Now factor the quadratic:

\[
(x - 4)(x + 2) = 0
\]

So the solutions are:

\[
x = 4 \quad \text{or} \quad x = -2
\]

Therefore, the horizontal intercepts are \( x = 4 \) and \( x = -2 \). These correspond to the points \( (4, 0) \) and \( (-2, 0) \).

---

### C. Range
Since the quadratic function opens upwards (as \( a = 3 > 0 \)), the minimum value occurs at the vertex.

The vertex is \( (1, -27) \), so the minimum value of \( y \) is \( -27 \). The range is all values of \( y \) starting from \( -27 \) and going to \( \infty \).

Thus, the range is:

\[
y \geq -27
\]

---

### D. Vertical Intercept
The vertical intercept is found by setting \( x = 0 \) in the quadratic function:

\[
y = 3(0)^2 - 6(0) - 24 = -24
\]

Thus, the vertical intercept is \( (0, -24) \).

---

### Summary:
A. **Vertex:** \( (1, -27) \)  
B. **Horizontal intercepts:** \( (4, 0) \) and \( (-2, 0) \)  
C. **Range:** \( y \geq -27 \)  
D. **Vertical intercept:** \( (0, -24) \)

---

Do you need any more details or have any further questions?

Here are 5 related questions for you to explore:
1. How would the graph change if the coefficient of \( x^2 \) were negative?
2. Can you derive the quadratic formula from completing the square?
3. How does the discriminant determine the number of real roots?
4. What is the significance of the axis of symmetry in a quadratic function?
5. How do you use quadratic equations in real-world applications?

**Tip:** When solving quadratic equations, always check whether they can be factored easily before applying the quadratic formula. It saves time!

For the quadratic function 𝑦 = 3𝑥^2 − 6𝑥 − 24, find the following:
A. Vertex
B. Horizontal intercepts
C. Range
D. Vertical Intercept

Discover how to find the vertex, horizontal intercepts, range, and vertical intercept of the quadratic function y = 3x^2 − 6x − 24. This detailed step-by-step guide explores key algebraic concepts, factoring, and graphing methods. Ideal for high school students (Grades 9-12).

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