Question

Identify the Key Characteristics of the Quadratic Function.
y=x^2-3x-4

Answer 1

To identify the key characteristics of the quadratic function \( y = x^2 - 3x - 4 \), we can analyze the function in several ways:

1. **Standard Form**: The function is in standard form \( y = ax^2 + bx + c \), where:
- \( a = 1 \) (the coefficient of \( x^2 \))
- \( b = -3 \)
- \( c = -4 \)

2. **Direction of the Parabola**: Since \( a > 0 \), the parabola opens upwards.

3. **Vertex**: The vertex of the quadratic function can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting in \( b = -3 \) and \( a = 1 \):
\[
x = -\frac{-3}{2 \cdot 1} = \frac{3}{2}
\]
To find the corresponding \( y \)-coordinate of the vertex, substitute \( x = \frac{3}{2} \) back into the original equation:
\[
y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 4 = \frac{9}{4} - \frac{9}{2} - 4 = \frac{9}{4} - \frac{18}{4} - \frac{16}{4} = \frac{9 - 18 - 16}{4} = \frac{-25}{4}
\]
Therefore, the vertex is at:
\[
\left(\frac{3}{2}, -\frac{25}{4}\right)
\]

4. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is given by:
\[
x = \frac{3}{2}
\]

5. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
y(0) = 0^2 - 3(0) - 4 = -4
\]
So, the y-intercept is at \( (0, -4) \).

6. **X-intercepts (Roots)**: The x-intercepts can be found by setting \( y = 0 \):
\[
0 = x^2 - 3x - 4
\]
Factoring the quadratic yields:
\[
(x - 4)(x + 1) = 0
\]
Thus, the x-intercepts are:
\[
x = 4 \quad \text{and} \quad x = -1
\]
So, the x-intercepts are \( (4, 0) \) and \( (-1, 0) \).

7. **Opening and Range**: Since the parabola opens upwards, the minimum value of the function (the y-coordinate of the vertex) is \( -\frac{25}{4} \). Therefore, the range of the function is:
\[
y \geq -\frac{25}{4}
\]

### Summary of Key Characteristics:
- **Direction**: Opens upwards
- **Vertex**: \( \left( \frac{3}{2}, -\frac{25}{4} \right) \)
- **Axis of Symmetry**: \( x = \frac{3}{2} \)
- **Y-intercept**: \( (0, -4) \)
- **X-intercepts**: \( (4, 0) \) and \( (-1, 0) \)
- **Range**: \( y \geq -\frac{25}{4} \)