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.