Questions: What are the key points on the graph of y=x^2-2x-120 ? Name the vertex, x-intercept(s); and y intercept. (1 point) x-intercepts: (-10,0)(12,0) y-intercept: (0,-120) vertex: (1,-121)

Solution Steps

To find the key points on the graph of the quadratic function \( y = x^2 - 2x - 120 \), we need to determine the vertex, the x-intercepts, and the y-intercept.

1. Vertex: The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Plug this x-value back into the equation to find the y-coordinate of the vertex.
2. x-intercepts: The x-intercepts are the points where the graph crosses the x-axis (i.e., where \( y = 0 \)). Solve the quadratic equation \( x^2 - 2x - 120 = 0 \) to find the x-values.
3. y-intercept: The y-intercept is the point where the graph crosses the y-axis (i.e., where \( x = 0 \)). Substitute \( x = 0 \) into the equation to find the y-value.

The vertex of the quadratic function \( y = x^2 - 2x - 120 \) is calculated using the formula for the x-coordinate of the vertex:

\[ x = -\frac{b}{2a} = -\frac{-2}{2 \cdot 1} = 1 \]

Substituting \( x = 1 \) back into the equation to find the y-coordinate:

\[ y = 1^2 - 2 \cdot 1 - 120 = 1 - 2 - 120 = -121 \]

Thus, the vertex is:

\[ \text{Vertex: } (1, -121) \]

The x-intercepts occur where \( y = 0 \). We solve the equation:

\[ x^2 - 2x - 120 = 0 \]

Factoring or using the quadratic formula, we find the x-intercepts:

\[ x = -10 \quad \text{and} \quad x = 12 \]

Thus, the x-intercepts are:

\[ \text{x-intercepts: } (-10, 0) \text{ and } (12, 0) \]

The y-intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the equation:

\[ y = 0^2 - 2 \cdot 0 - 120 = -120 \]

Thus, the y-intercept is:

\[ \text{y-intercept: } (0, -120) \]

Final Answer