Maths 1 Week 3 Summary

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Quadratic Functions Summary

A quadratic function is a polynomial function of degree 2, generally represented as \( f(x) = ax^2 + bx + c \), where a, b, and c are constants, and a \neq 0.

Graph of Quadratic Function

The graph of a quadratic function is a parabola. It can open either upward or downward depending on the sign of the coefficient a. If a > 0, the parabola opens upward. If a < 0, it opens downward.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror images. It can be found using the formula \( x = -\frac{b}{2a} \).

Coordinates of the Vertex

The vertex of the parabola is the highest or lowest point on the graph, depending on whether it opens downward or upward. The coordinates of the vertex can be found using the formula \( (x, y) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \).

Parabola Opening Upward and Downward

If a > 0, the parabola opens upward, and the vertex represents the minimum point. If a < 0, the parabola opens downward, and the vertex represents the maximum point.

Maximum and Minimum Value of Parabola

The maximum or minimum value of the parabola is the y-coordinate of the vertex. For a > 0, the minimum value is \( f\left(-\frac{b}{2a}\right) \). For a < 0, the maximum value is \( f\left(-\frac{b}{2a}\right) \).

Range of Parabola

The range of a parabola depends on its direction. For a > 0, the range is \( [f\left(-\frac{b}{2a}\right), \infty) \). For a < 0, the range is \( (-\infty, f\left(-\frac{b}{2a}\right)] \).

Slope of Quadratic Function

The slope of a quadratic function is not constant. It changes at different points along the curve. The slope at any point x can be found using the derivative \( f'(x) = 2ax + b \).

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). The solutions to this equation are called the roots or zeros of the quadratic function.

Difference Between Equation and Function

A quadratic function represents a relationship between x and y values, while a quadratic equation is a statement that the quadratic function equals zero.

Difference Between Roots and Zeros

The roots or zeros of a quadratic function are the values of x that make the function equal to zero. They are the solutions to the quadratic equation \( ax^2 + bx + c = 0 \).

Finding Roots Using Discriminant Formula

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). The discriminant determines the nature of the roots:

• If D > 0, there are two distinct real roots.
• If D = 0, there is one real root (a repeated root).
• If D < 0, there are no real roots (the roots are complex).

GA