Given the following parabolas, find the vertex, x-intercepts, and y-intercepts.
• y = 2x2 - 8x - 7
• y = - 16x2 + 120x + 9
• y = -0.2x2 + 4.6х - 120

Let's solve each parabola step-by-step by finding the **vertex**, **x-intercepts**, and **y-intercepts**.

### 1. **For** \( y = 2x^2 - 8x - 7 \)

#### Vertex
The formula for the x-coordinate of the vertex is:
\[
x = -\frac{b}{2a}
\]
where \( a = 2 \) and \( b = -8 \). Substituting these values:
\[
x = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2
\]
To find the y-coordinate, substitute \( x = 2 \) back into the equation:
\[
y = 2(2)^2 - 8(2) - 7 = 2 \cdot 4 - 16 - 7 = 8 - 16 - 7 = -15
\]
So, the vertex is at \( (2, -15) \).

#### x-Intercepts
To find the x-intercepts, set \( y = 0 \):
\[
2x^2 - 8x - 7 = 0
\]
Using the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2} = \frac{8 \pm \sqrt{64 + 56}}{4} = \frac{8 \pm \sqrt{120}}{4} = \frac{8 \pm 2\sqrt{30}}{4} = 2 \pm \frac{\sqrt{30}}{2}
\]
Thus, the x-intercepts are \( x = 2 + \frac{\sqrt{30}}{2} \) and \( x = 2 - \frac{\sqrt{30}}{2} \).

#### y-Intercept
For the y-intercept, set \( x = 0 \):
\[
y = 2(0)^2 - 8(0) - 7 = -7
\]
So, the y-intercept is at \( (0, -7) \).

---

### 2. **For** \( y = -16x^2 + 120x + 9 \)

#### Vertex
Using \( x = -\frac{b}{2a} \):
\[
x = -\frac{120}{2 \cdot -16} = \frac{120}{32} = 3.75
\]
Substitute \( x = 3.75 \) back into the equation for \( y \):
\[
y = -16(3.75)^2 + 120(3.75) + 9 = -16 \cdot 14.0625 + 450 + 9 = -225 + 450 + 9 = 234
\]
So, the vertex is at \( (3.75, 234) \).

#### x-Intercepts
Set \( y = 0 \):
\[
-16x^2 + 120x + 9 = 0
\]
Using the quadratic formula:
\[
x = \frac{-120 \pm \sqrt{120^2 - 4 \cdot (-16) \cdot 9}}{2 \cdot -16} = \frac{120 \pm \sqrt{14400 + 576}}{-32} = \frac{120 \pm \sqrt{14976}}{-32}
\]
Simplifying further will provide the exact x-intercepts.

#### y-Intercept
Set \( x = 0 \):
\[
y = -16(0)^2 + 120(0) + 9 = 9
\]
So, the y-intercept is \( (0, 9) \).

---

### 3. **For** \( y = -0.2x^2 + 4.6x - 120 \)

#### Vertex
Using \( x = -\frac{b}{2a} \):
\[
x = -\frac{4.6}{2 \cdot -0.2} = \frac{4.6}{

Given the following parabolas, find the vertex, x-intercepts, and y-intercepts.
• y = 2x^2 - 8x - 7
• y = -16x^2 + 120x + 9
• y = -0.2x^2 + 4.6x - 120

Learn how to find the vertex, x-intercepts, and y-intercepts of quadratic functions. This example covers parabolas such as y = 2x^2 - 8x - 7, using key algebraic concepts and formulas like the vertex and quadratic formulas. Ideal for students in Grades 9-10.

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