Problem 63 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. $$y=\frac{1}{2} x^{2}$$

Step-by-Step Solution

Answer

The graph of the quadratic function \(y = \frac{1}{2}x^{2}\) is a parabola opening upwards, with its vertex and root at the origin (0,0).

Step 1: Understand the Function

The given function is a quadratic function in the form \(f(x) = ax^{2}\). Here, the coefficient \(a\) is \(\frac{1}{2}\). Since \(a>0\), the parabola opens upwards.

Step 2: Find the Vertex

For a function in the form \(f(x) = ax^{2}\), the vertex is always at the origin, (0,0). Thus, our vertex point is (0,0).

Step 3: Find the Roots

To find the roots, set \(y = 0\) and solve for \(x\). Here, we solve \(\frac{1}{2}x^{2} = 0\). There's one root for this equation, \(x = 0\). So, our root is (0,0).

Step 4: Sketch the graph

Sketch the graph by marking the vertex and roots on a coordinate plane. Because the parabola is symmetric, plot points for both positive and negative values of x. The points will lie on a curved path, forming a U-shaped graph that opens upwards. The graph intersects the y-axis at the origin.