Section 8.2: Graphing Quadratic Functions
Graphing a quadratic function involves identifying the vertex, axis of symmetry, direction of opening, and intercepts.
Vertex form: \( f(x) = a(x-h)^2 + k \) makes it easy to graph: vertex at \( (h, k) \), axis of symmetry \( x = h \), opens up if \( a>0 \), down if \( a<0 \).
Example 1
Graph \( f(x) = (x-2)^2 + 3 \).
Vertex at \( (2,3) \)
Axis of symmetry: \( x = 2 \)
Opens upwards since \( a = 1 > 0 \)
y-intercept: \( f(0) = (0-2)^2 + 3 = 7 \)
Plot vertex and a few points on either side of the axis to sketch the parabola.
Example 2
Graph \( f(x) = -2(x+1)^2 + 4 \).
Vertex: \( (-1, 4) \)
Axis of symmetry: \( x = -1 \)
Opens downward since \( a = -2 < 0 \)
y-intercept: \( f(0) = -2(1)^2 + 4 = 2 \)
Sketch parabola using vertex, axis, and intercept.
Practice Problems
- Graph \( f(x) = x^2 - 6x + 8 \) by converting to vertex form.
- Graph \( f(x) = -x^2 + 4x + 5 \) and label the vertex.
- Find the vertex and axis of symmetry for \( f(x) = 2x^2 - 8x + 6 \).
- Sketch \( f(x) = 3(x+2)^2 - 1 \) showing vertex and a few points.
- Graph \( f(x) = -(x-3)^2 + 2 \) and determine y-intercept.