Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as the second power of x. (y=kx^2)

Verified step by step guidance

1. Identify the type of variation described: y varies directly as the second power of x means the equation can be written as \(y = kx^{2}\), where \(k > 0\).
2. Recognize the shape of the graph for \(y = kx^{2}\): since \(k\) is positive and the power of \(x\) is 2, the graph is a parabola opening upwards.
3. Recall that the graph of \(y = kx^{2}\) is symmetric about the y-axis because the exponent on \(x\) is even.
4. Eliminate any graphs that do not show a parabola opening upwards or are not symmetric about the y-axis.
5. Match the statement to the graph that shows a parabola opening upwards with vertex at the origin, which corresponds to the equation \(y = kx^{2}\) with \(k > 0\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is equal to a constant multiplied by another variable raised to a power. In this case, y varies directly as x squared, meaning y = kx², where k is a positive constant. This implies that as x increases, y changes proportionally to the square of x.

Quadratic Functions and Their Graphs

A quadratic function has the form y = ax² + bx + c, and its graph is a parabola. When y = kx² with k > 0, the parabola opens upward and is symmetric about the y-axis. The shape and steepness depend on the value of k, with larger k values making the parabola narrower.

Interpreting Graphs of Power Functions

Graphs of power functions y = kx^n show how y changes with x raised to a power n. For n = 2, the graph is a parabola. Understanding how the exponent affects the curve helps match equations to their graphs, especially recognizing the shape and direction based on the sign and magnitude of k.