Answer:
Here's the breakdown for the equation y = 2x² - 5x + 2:
X Intercepts:
* Method 1: Factoring
The x intercepts occur when y = 0. So you need to solve the equation:
0 = 2x² - 5x + 2
Factoring the expression: (2x - 1)(x - 2) = 0
This gives you two possible values for x: x = 1/2 and x = 2.
* Alternate Method 2: The Quadratic Formula
You can use the quadratic formula to solve for the x-intercepts directly. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 2, b = -5, and c = 2. Plugging these values into the formula will give you the same solutions as factoring (x = 1/2 and x = 2).
Vertex:
* Analyze the equation: The equation is in the form y = a(x - h)² + k, where (h, k) is the vertex. In this case, a = 2 (positive), so the vertex represents a minimum point.
Steps to Graph:
* Plot the x-intercepts: These are the points where the line crosses the x-axis, so they are (1/2, 0) and (2, 0).
* Consider the vertex: The vertex is (h, k), which in this case is not directly on the x-axis (since a ≠ 0). However, knowing it's a minimum point tells you the graph curves downwards around this point.
* Use symmetry: Since the equation is in the form y = a(x - h)² + k, the graph is symmetrical around the vertical line x = h (which is x = 1/2 in this case). You can plot a point based on one side of the vertex and mirror it to the other side.
By following these steps and using the symmetry property, you can effectively graph the equation.
