To graph the function [tex]\( f(x) = \frac{1}{2} x^2 \)[/tex], follow these steps:
### Step 1: Identify Key Features of the Function
The function [tex]\( f(x) = \frac{1}{2} x^2 \)[/tex] is a quadratic function and its graph is a parabola. Here are some key features:
- Vertex: The vertex of the parabola for this quadratic equation is at the origin [tex]\((0, 0)\)[/tex].
- Axis of Symmetry: The axis of symmetry is the y-axis or the line [tex]\( x = 0 \)[/tex].
- Direction: Since the coefficient of [tex]\( x^2 \)[/tex] is positive ([tex]\( \frac{1}{2} > 0 \)[/tex]), the parabola opens upwards.
### Step 2: Generate Data Points
To graph the function, evaluate [tex]\( f(x) \)[/tex] at several values of [tex]\( x \)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( f(-10) = \frac{1}{2}(-10)^2 = 50 \)[/tex].
- When [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = \frac{1}{2}(-5)^2 = 12.5 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = \frac{1}{2}(0)^2 = 0 \)[/tex].
- When [tex]\( x = 5 \)[/tex], [tex]\( f(5) = \frac{1}{2}(5)^2 = 12.5 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( f(10) = \frac{1}{2}(10)^2 = 50 \)[/tex].
These points provide a general structure of the parabola.
### Step 3: Plot the Points and Sketch the Parabola
Plot the points [tex]\((-10, 50)\)[/tex], [tex]\((-5, 12.5)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((5, 12.5)\)[/tex], and [tex]\((10, 50)\)[/tex] on a coordinate plane. Then draw a smooth curve through these points to form the parabola. The graph should look symmetric about the y-axis, with the vertex at the origin.
### Step 4: Determine the Domain and Range
- Domain: The domain of [tex]\( f(x) \)[/tex] refers to all possible values of [tex]\( x \)[/tex]. Since [tex]\( x \)[/tex] can take any real number value, the domain is:
[tex]\[
(-\infty, \infty)
\][/tex]
- Range: The range of [tex]\( f(x) \)[/tex] refers to all possible values of [tex]\( f(x) \)[/tex]. Since the parabola opens upwards and the vertex is at the lowest point [tex]\( (0, 0) \)[/tex], [tex]\( f(x) \)[/tex] can only take non-negative values. Thus, the range is:
[tex]\[
[0, \infty)
\][/tex]
### Summary
The function [tex]\( f(x) = \frac{1}{2} x^2 \)[/tex] has a domain of [tex]\( (-\infty, \infty) \)[/tex] and a range of [tex]\( [0, \infty) \)[/tex]. The graph is a parabolic curve opening upwards with its vertex at the origin.
