To sketch the graph of the function [tex]\( f(x) = x^2 - 4 \)[/tex] for the interval [tex]\(-2 \leq x \leq 4\)[/tex], follow these steps:
### Step 1: Determine Key Features of the Function
1. Standard Form: The function [tex]\( f(x) = x^2 - 4 \)[/tex] is a quadratic function in standard form, [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -4 \)[/tex].
2. Vertex: The vertex form of a quadratic function is [tex]\( f(x) = a(x-h)^2 + k \)[/tex]. For this function, the vertex is at [tex]\( (0, -4) \)[/tex].
3. Axis of Symmetry: The axis of symmetry for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is defined by [tex]\( x = -\frac{b}{2a} \)[/tex]. Here, [tex]\( b = 0 \)[/tex], so [tex]\( x = 0 \)[/tex].
4. Y-Intercept: The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex], we get:
[tex]\[
f(0) = 0^2 - 4 = -4
\][/tex]
5. X-Intercepts: The x-intercepts occur when [tex]\( f(x) = 0 \)[/tex]. Solving for [tex]\( x \)[/tex], we have:
[tex]\[
0 = x^2 - 4 \implies x^2 = 4 \implies x = \pm 2
\][/tex]
### Step 2: Create a Table of Values
Calculate the function values within the given interval [tex]\(-2 \leq x \leq 4\)[/tex]:
[tex]\[
\begin{array}{c|c}
x & f(x) \\
\hline
-2 & (-2)^2 - 4 = 0 \\
-1 & (-1)^2 - 4 = -3 \\
0 & 0^2 - 4 = -4 \\
1 & 1^2 - 4 = -3 \\
2 & 2^2 - 4 = 0 \\
3 & 3^2 - 4 = 5 \\
4 & 4^2 - 4 = 12 \\
\end{array}
\][/tex]
### Step 3: Plot the Points
Plot each calculated point on the coordinate grid:
- [tex]\((-2, 0)\)[/tex]
- [tex]\((-1, -3)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- [tex]\((1, -3)\)[/tex]
- [tex]\((2, 0)\)[/tex]
- [tex]\((3, 5)\)[/tex]
- [tex]\((4, 12)\)[/tex]
### Step 4: Sketch the Curve
Draw a smooth parabolic curve through these points, ensuring it is symmetrical with respect to the y-axis (x = 0), and opening upwards.
### Step 5: Mark Features
- Vertex: [tex]\( (0, -4) \)[/tex]
- X-Intercepts: [tex]\( (-2, 0) \)[/tex] and [tex]\( (2, 0) \)[/tex]
- Y-Intercept: [tex]\( (0, -4) \)[/tex]
- Interval: [tex]\(-2 \leq x \leq 4\)[/tex]
### Graph of [tex]\( f(x) = x^2 - 4 \)[/tex]
The parabolic curve will start from the point [tex]\((-2, 0)\)[/tex], dip down to the vertex [tex]\((0, -4)\)[/tex], and rise up again passing through [tex]\( (2, 0) \)[/tex], extending upward to [tex]\( (4, 12) \)[/tex].
This completes the sketch of the graph of the quadratic function [tex]\( f(x) = x^2 - 4 \)[/tex] over the interval [tex]\(-2 \leq x \leq 4\)[/tex].
