Identify the vertex of the quadratic function \( g(x) = 4(x - 1)^2 - 1 \).

The vertex is at \( (1, -1) \).

Find the x-coordinate of the vertex for \( f(x) = x^2 + 2x \).

\( h = -b/2a = -2/(2(1)) = -1 \).

What are the properties of a quadratic function in vertex form?

A function in vertex form \( f(x) = a(x-h)^2 + k \) has a vertex at \( (h, k) \) and an axis of symmetry at \( x = h \). It opens up if \( a > 0 \) and down if \( a < 0 \).

Graphing Quadratic Functions: Vertex Form, Standard Form and Examples

Quadratic functions are polynomial functions of degree 2. Their graphs are parabolas. We will explore the basic function \( f(x) = x^2 \), the transformed vertex form \( f(x) = a(x - h)^2 + k \), and the standard form \( f(x) = ax^2 + bx + c \).

1. The Basic Parabola: \( f(x) = x^2 \)

Before graphing transformations, we analyze the parent function. We use a table of values to plot key points.

\( x \)	\( f(x) = x^2 \)
-3	9
-2	4
-1	1
0	0 (Vertex)
1	1
2	4
3	9

graph of basic quadratic function f(x) = x^2

2. Properties of the Vertex Form: \( g(x) = a(x - h)^2 + k \)

The Vertex Form allows for immediate identification of the parabola's center and orientation:

Vertex: Located at the point \( (h, k) \).
Axis of Symmetry: The vertical line \( x = h \).
Direction of Opening:
- If \( a > 0 \): Opens upward (Vertex is a Minimum).
- If \( a < 0 \): Opens downward (Vertex is a Maximum).
Range:
- \( y \ge k \) if \( a > 0 \).
- \( y \le k \) if \( a < 0 \).

Example 2: Graphing \( g(x) = 4(x - 1)^2 - 1 \)

View Step-by-Step Solution

1. Identify Parameters: \( a=4, h=1, k=-1 \). Vertex is (1, -1).

2. Intercepts:
y-intercept: \( g(0) = 4(0-1)^2 - 1 = 3 \). Point: (0, 3).
x-intercepts: Solve \( 4(x-1)^2 - 1 = 0 \). Points: (0.5, 0) and (1.5, 0).

graph of quadratic function g(x) = 4(x-1)^2 - 1

3. Graphing in Standard Form: \( f(x) = ax^2 + bx + c \)

To graph standard form, we calculate the vertex coordinates \( (h, k) \) first:

\[ h = -\frac{b}{2a} \quad \text{and} \quad k = f(h) \]

Example 5: Graphing \( p(x) = -2x^2 + 3x + 1 \)

Find the vertex, intercepts, and sketch the graph.

View Solution

1. Vertex:
\( h = -3 / (2 \cdot -2) = 0.75 \)
\( k = p(0.75) = -2(0.75)^2 + 3(0.75) + 1 = 2.125 \). Vertex: (0.75, 2.13).

2. Graph: Since \( a = -2 \), the parabola opens downward.

Graphing Quadratic Functions

1. The Basic Parabola: \( f(x) = x^2 \)

2. Properties of the Vertex Form: \( g(x) = a(x - h)^2 + k \)

Example 2: Graphing \( g(x) = 4(x - 1)^2 - 1 \)

3. Graphing in Standard Form: \( f(x) = ax^2 + bx + c \)

Example 5: Graphing \( p(x) = -2x^2 + 3x + 1 \)

Continue Practice