Match the quadratic function with its graph.
f(x) = x2 − 3









To match the quadratic function \( f(x) = x^2 - 3 \) with its graph, we need to understand the following properties:

1. **Vertex**: The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). In this case, \( a = 1 \), \( b = 0 \), and \( c = -3 \). This means the vertex of the parabola is at \( (0, -3) \).
2. **Shape**: Since \( a = 1 \) (positive), the parabola opens **upward**.
3. **Axis of symmetry**: The axis of symmetry is the vertical line \( x = 0 \), because there is no \( x \)-term (i.e., \( b = 0 \)).
4. **Y-intercept**: The y-intercept occurs when \( x = 0 \). Plugging \( x = 0 \) into \( f(x) = x^2 - 3 \), we get \( f(0) = -3 \). So the parabola crosses the y-axis at \( (0, -3) \).
5. **X-intercepts**: Solve \( x^2 - 3 = 0 \) to find the x-intercepts. This gives:
   \[
   x^2 = 3 \quad \Rightarrow \quad x = \pm \sqrt{3} \approx \pm 1.732.
   \]
   So the parabola crosses the x-axis at \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).

### Key Points for the Graph:
- Vertex: \( (0, -3) \)
- Opens upward
- Y-intercept at \( (0, -3) \)
- X-intercepts at approximately \( (1.732, 0) \) and \( (-1.732, 0) \)

If you have a set of graphs to choose from, the correct graph will be a parabola opening upwards with its vertex at \( (0, -3) \) and crossing the x-axis at points near \( \pm 1.732 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the vertex of a quadratic function?
2. What is the role of the coefficient \( a \) in determining the shape of a parabola?
3. How can we determine the axis of symmetry for a quadratic function?
4. What does the discriminant tell us about the x-intercepts of a quadratic function?
5. How does the y-intercept of a quadratic function relate to the constant \( c \)?

### Tip:
The vertex form of a quadratic equation \( f(x) = a(x - h)^2 + k \) makes it easy to identify the vertex directly.

Match the quadratic function with its graph. f(x) = x^2 − 3

Match the quadratic function f(x) = x^2 - 3 to its graph by analyzing the vertex, axis of symmetry, and intercepts. Understand key concepts like parabolas, the quadratic equation, and the role of coefficients in shaping the graph. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation