How to Graph A X-2 2 Without A Calculator

Understanding the Function

The general form of a quadratic function is f(x) = a(x - h)² + k, where:

• (h, k) is the vertex of the parabola
• a determines the parabola's direction and width

For the function a x-2 2, we can rewrite it in vertex form:

This shows the vertex is at (2, 2). The value of 'a' affects the parabola's shape:

• If a > 0, the parabola opens upwards
• If a < 0, the parabola opens downwards
• The larger |a| is, the narrower the parabola

Graphing Steps

Step 1: Identify the Vertex

From the vertex form, the vertex is at (2, 2). Plot this point on the graph.

Step 2: Determine the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. For our function, it's x = 2.

Step 3: Find Additional Points

Choose x-values on either side of the vertex to find corresponding y-values. A good strategy is to use x = 0, x = 1, x = 3, and x = 4.

Step 4: Plot the Points

Plot each (x, y) point on the coordinate plane. Connect the points with a smooth curve to form the parabola.

Step 5: Sketch the Graph

Draw the parabola based on the plotted points, ensuring it passes through the vertex and is symmetric about the axis of symmetry.

Key Points

For a x-2 2, the key points include:

• Vertex: (2, 2)
• Y-intercept: Set x = 0 → f(0) = a(0 - 2)² + 2 = 4a + 2
• X-intercepts: Set y = 0 → a(x - 2)² + 2 = 0 → (x - 2)² = -2/a → x = 2 ± √(-2/a)

Example

Let's graph f(x) = -2(x - 2)² + 2:

1. Vertex at (2, 2)
2. Y-intercept: f(0) = -2(0 - 2)² + 2 = -8 + 2 = -6 → (0, -6)
3. X-intercepts: -2(x - 2)² + 2 = 0 → (x - 2)² = -1 → x = 2 ± i → No real x-intercepts

Plot these points and draw the parabola opening downward with vertex at (2, 2).