Understanding Quadratic Functions

Front

What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

What are the standard forms of a quadratic function?

Back

The standard forms are: 1) Standard form: f(x) = ax² + bx + c; 2) Vertex form: f(x) = a(x-h)² + k; 3) Factored form: f(x) = a(x-r₁)(x-r₂), where r₁ and r₂ are the roots.

Front

What are the x-intercepts of a quadratic function?

Back

The x-intercepts, also known as the zeros or roots, are the points where the graph of the function intersects the x-axis, found by solving f(x) = 0.

Front

How do you find the vertex of a quadratic function in standard form?

Back

The vertex can be found using the formula: h = -b/(2a) and k = f(h), where (h, k) is the vertex.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation x = h, where h is the x-coordinate of the vertex.

Front

What does the discriminant of a quadratic equation tell us?

Back

The discriminant (D = b² - 4ac) indicates the nature of the roots: D > 0 means two distinct real roots, D = 0 means one real root, and D < 0 means no real roots.

Front

How do you solve a quadratic equation by factoring?

Back

To solve by factoring, rewrite the equation in the form ax² + bx + c = 0, factor it into (px + q)(rx + s) = 0, and set each factor to zero.

Front

What is the vertex form of a quadratic function?

Back

The vertex form is f(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola.

Front

What is the significance of the 'a' value in a quadratic function?

Back

The 'a' value determines the direction of the parabola (upward if a > 0, downward if a < 0) and affects the width of the parabola.

Front

What are complex roots in a quadratic equation?

Back

Complex roots occur when the discriminant is negative, resulting in roots of the form a ± bi, where a and b are real numbers.