Section 8.1: Introduction to Quadratics
A quadratic function is a polynomial function of degree 2 and can be written in the standard form:
\( f(x) = ax^2 + bx + c \), where \( a \neq 0 \).
The graph of a quadratic function is a parabola, which opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
Example 1
Identify the coefficients and the direction of the parabola for \( f(x) = 2x^2 - 4x + 1 \).
Coefficient \( a = 2 \), \( b = -4 \), \( c = 1 \).
Since \( a > 0 \), the parabola opens upwards.
Example 2
Determine the vertex of \( f(x) = -x^2 + 6x - 5 \).
Vertex formula: \( x_v = -\frac{b}{2a} = -\frac{6}{2(-1)} = 3 \)
\( y_v = f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4 \)
Vertex: \( (3, 4) \)
Practice Problems
- Write the standard form of a quadratic with \( a = 1 \), \( b = -5 \), \( c = 6 \).
- For \( f(x) = -3x^2 + 12x - 7 \), identify the vertex and direction of the parabola.
- Graph \( f(x) = x^2 - 4x + 3 \) and label the vertex.
- Determine the zeros of \( f(x) = 2x^2 - 8x + 6 \).
- Describe the transformations of \( f(x) = -(x-2)^2 + 5 \) compared to \( f(x) = x^2 \).