Question

graph the equation.
$y = \frac{1}{4}x^2 - 2x$

Explanation:

Step1: Identify the parabola's properties

The equation is \( y = \frac{1}{4}x^2 - 2x \), a quadratic (parabola). The coefficient of \( x^2 \) is positive, so it opens upward. To find the vertex, we can complete the square or use \( x = -\frac{b}{2a} \). Here, \( a = \frac{1}{4} \), \( b = -2 \). So \( x = -\frac{-2}{2 \times \frac{1}{4}} = \frac{2}{\frac{1}{2}} = 4 \). Then \( y = \frac{1}{4}(4)^2 - 2(4) = 4 - 8 = -4 \). Wait, but the graph shown has a vertex? Wait, maybe the graph is incorrect? Wait, no, let's check points. Let's plug \( x = 0 \): \( y = 0 \). \( x = 4 \): \( y = \frac{1}{4}(16) - 8 = 4 - 8 = -4 \). But the graph shown has vertex at (0,0)? Wait, maybe the given graph is not correct, but the problem is to graph \( y = \frac{1}{4}x^2 - 2x \). Let's find another point: \( x = 8 \), \( y = \frac{1}{4}(64) - 16 = 16 - 16 = 0 \). So roots at \( x = 0 \) and \( x = 8 \), vertex at (4, -4). But the provided graph has a vertex at (0,0) and a point at (5,5). Wait, maybe the question is to check if the graph matches? Wait, the problem says "Graph the equation" but the graph is shown. Maybe the task is to identify the correct graph or analyze? Wait, perhaps the original problem is to see if the given graph is correct. Let's check the point (5,5): plug \( x = 5 \) into the equation: \( y = \frac{1}{4}(25) - 10 = \frac{25}{4} - 10 = 6.25 - 10 = -3.75 \), not 5. So the given graph is incorrect. But maybe the problem is to graph it correctly. Let's proceed to graph it properly.

Step2: Find key points

• Roots: Set \( y = 0 \): \( \frac{1}{4}x^2 - 2x = 0 \) → \( x(\frac{1}{4}x - 2) = 0 \) → \( x = 0 \) or \( \frac{1}{4}x - 2 = 0 \) → \( x = 8 \). So points (0,0) and (8,0).
• Vertex: At \( x = 4 \), \( y = -4 \), so (4, -4).
• Another point: \( x = 4 \), \( y = -4 \); \( x = 2 \), \( y = \frac{1}{4}(4) - 4 = 1 - 4 = -3 \); \( x = 6 \), \( y = \frac{1}{4}(36) - 12 = 9 - 12 = -3 \).

So the correct graph should have roots at (0,0) and (8,0), vertex at (4, -4), opening upward. The given graph in the image has a vertex at (0,0) and a point (5,5) which is not on the curve, so it's incorrect. But if we were to graph it, we plot (0,0), (8,0), (4, -4), and other points like (2, -3), (6, -3).

Answer:

The correct graph of \( y = \frac{1}{4}x^2 - 2x \) has roots at \( (0, 0) \) and \( (8, 0) \), vertex at \( (4, -4) \), and opens upward. The provided graph is incorrect; the correct graph should be drawn with these key points.