Precalc unit 3 notes

Vertex

The highest or lowest point of a parabola, found using the formula x = -b/2a.

Axis of Symmetry

An invisible line that cuts a parabola in half, passing through the vertex.

Quadratic Function

An equation in the form y = ax² + bx + c, which graphs as a parabola.

Parabola

A U-shaped curve that represents a quadratic function.

Y-Intercept

The point where the graph crosses the y-axis (set x = 0).

X-Intercepts

The points where the graph crosses the x-axis (set y = 0).

Maximum Value

If the parabola opens downward, the vertex gives the highest value.

Minimum Value

If the parabola opens upward, the vertex gives the lowest value.

Vertex Form

The equation y = a(x - h)² + k, where (h, k) is the vertex.

Standard Form

The equation y = ax² + bx + c, used for finding the axis of symmetry.

Solving Quadratic Inequalities

Graph the parabola, then shade above or below depending on the inequality sign.

Dashed vs. Solid Line

Use a dashed line for < or >; use a solid line for ≤ or ≥.

How to find the vertex using the quadratic formula?

Use the formula x = -b/2a to find the x-coordinate of the vertex, then plug this value back into the quadratic function to find the y-coordinate.

Example of finding x-intercepts.

To find the x-intercepts, set y = 0 in the quadratic equation ax² + bx + c = 0, and solve for x using factoring, completing the square, or the quadratic formula.

Example of determining the y-intercept.

To find the y-intercept, substitute x = 0 into the quadratic equation. The result gives the point where the graph crosses the y-axis.

How to graph a parabola given the vertex?

Plot the vertex, determine the axis of symmetry, find additional points by substituting x-values, and sketch the U-shaped curve.

How to solve quadratic inequalities using a graph?

Graph the corresponding quadratic function, identify the x-intercepts, and then shade above or below the parabola according to the inequality sign.