5.7&5.8 HW Determine Quadratic Function and Systems

Front

What is a quadratic function?

Back

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

Front

What does it mean for a quadratic function to have one root?

Back

A quadratic function has one root when its graph touches the x-axis at exactly one point, indicating that the discriminant (b^2 - 4ac) is equal to zero.

Front

How do you find the roots of a quadratic function?

Back

The roots of a quadratic function can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Front

What is the vertex of a quadratic function?

Back

The vertex of a quadratic function is the highest or lowest point on the graph, located at the point (h, k) where h = -b/(2a) and k = f(h).

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 = -b/(2a).

Front

What is a system of equations?

Back

A system of equations is a set of two or more equations with the same variables, which can be solved simultaneously to find the values of the variables.

Front

What does it mean for a system of equations to have no solutions?

Back

A system of equations has no solutions when the equations represent parallel lines that never intersect.

Front

What is the discriminant of a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b^2 - 4ac, which determines the nature of the roots.

Front

What are the possible values of the discriminant and their meanings?

Back

If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; if it is negative, there are two complex roots.

Front

How do you solve a quadratic equation by completing the square?

Back

To solve by completing the square, rearrange the equation to isolate the constant, then add the square of half the coefficient of x to both sides, and factor.