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# 10.1: Quadratic Functions and Their Graphs

Difficulty Level: Basic Created by: CK-12
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Practice Quadratic Functions and Their Graphs

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$x$-intercept of a parabola

As with linear equations, the $x$-intercepts of a parabola are where the graph intersects the $x$-axis. The $y$-value is zero at the $x$-intercepts.

The variable $a$ in the equation $y=ax^2+bx+c$ is called the leading coefficient of the quadratic equation.

minimums and maximums of a parabola

An equation of the form $y=ax^2+bx+c$ forms a parabola. If $a$ is positive, the parabola will open upward. The vertex will be a minimum. If $a$ is negative, the parabola will open downward. The vertex will be a maximum.

symmetry of a parabola

A parabola can be divided in half by a vertical line. Because of this, parabolas have symmetry. The vertical line dividing the parabola into two equal portions is called the line of symmetry.

vertex of a parabola

All parabolas have a vertex, the ordered pair that represents the bottom (or the top) of the curve. The line of symmetry always goes through the vertex. The vertex of a parabola is the ordered pair $(h, k)$.

Coefficient

A coefficient is the number in front of a variable.

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.

domain

The domain of a function is the set of $x$-values for which the function is defined.

Horizontal shift

A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right.

Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".

A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.

standard form

The standard form of a quadratic function is $f(x)=ax^{2}+bx+c$.

Symmetry

A figure has symmetry if it can be transformed and still look the same.

Vertex

The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.

vertical axis

The vertical axis is also referred to as the $y$-axis of a coordinate graph. By convention, we graph the output variable on the $y$-axis.

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