Algebra 1 - Quadratic Functions

Introduction

• A quadratic function is a function whose defining equation can be expressed in the form $\mathit{f}\left(x\right)\mathbf{=}\mathit{a}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{x}\mathbf{+}\mathit{c}\mathbf{ }\mathit{o}\mathit{r}\mathbf{ }\mathit{y}\mathbf{=}\mathit{a}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{x}\mathbf{+}\mathit{c}\mathbf{,}\mathbf{ }\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{a}\mathbf{,}\mathbf{ }\mathit{b}\mathbf{,}\mathbf{ }\mathit{c}\mathbf{\in }\mathbf{ℝ}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{a}\mathbf{\ne }\mathbf{0}$. 
• A quadratic function can be written in a standard form $\mathit{y}\mathbf{=}\mathit{a}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{x}\mathbf{+}\mathit{c}$, or the vertex form $\mathit{y}\mathbf{=}\mathit{a}{\left(x-h\right)}^{\mathbf{2}}\mathbf{+}\mathit{k}\mathbf{,}\mathbf{ }\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\left(h,k\right)$ are the coordinates of the vertex. 
• The vertex $\left(h,k\right)$ can be solved by using the following formulas:

    $\mathit{h}\mathbf{=}\mathbf{-}\frac{\mathbf{b}}{\mathbf{2}\mathbf{a}}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{k}\mathbf{=}\frac{\mathbf{4}\mathbf{a}\mathbf{c}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}}{\mathbf{4}\mathbf{a}}$

• The minimum or maximum value of a quadratic function is determined by using the formula $\mathit{k}\mathbf{=}\frac{\mathbf{4}\mathbf{a}\mathbf{c}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}}{\mathbf{4}\mathbf{a}}$. Another way is by directly substituting the value of $\mathit{h}$ to the function and computing for the function value. The graphical method can also determine a quadratic function's minimum or maximum point. 
• The graph of a quadratic function is a smooth u-shaped curve called a parabola. 

Forms of Quadratic Function

The following are forms of a quadratic function:

1. $\mathit{y}\mathbf{=}\mathbf{\pm }\mathit{a}{\mathit{x}}^{\mathbf{2}}\mathbf{,}\mathbf{ }\mathit{v}\mathit{e}\mathit{r}\mathit{t}\mathit{e}\mathit{x}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\left(0,0\right)$

Example: Graph of $f\left(x\right)=\frac{1}{4}{x}^2 and g\left(x\right)=-2x^2$

2. $\mathit{y}\mathbf{=}\mathbf{\pm }\mathit{a}{\mathit{x}}^{\mathbf{2}}\mathbf{\pm }\mathit{k}\mathbf{,}\mathbf{ }\mathit{v}\mathit{e}\mathit{r}\mathit{t}\mathit{e}\mathit{x}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{k}\mathbf{)}$. The graph of the function has a vertical shift. The parabola shifts upward if $\mathit{k}\mathbf{>}\mathbf{0}$ or it shifts downward if $\mathit{k}\mathbf{<}\mathbf{0}$. The vertex of this parabola lies either on the positive y-axis or the negative y-axis.

Example: Graph of $f\left(x\right)=-x^2-1 \left(red\right), g\left(x\right)=x^2-4 \left(orange\right), and h\left(x\right)=x^2+2 \left(blue\right)$

3. $\mathit{y}\mathbf{=}\mathbf{\pm }\mathit{a}{\left(x-h\right)}^{\mathbf{2}}\mathbf{,}\mathbf{ }\mathit{v}\mathit{e}\mathit{r}\mathit{t}\mathit{e}\mathit{x}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\left(h,0\right)$. The graph of the function has a horizontal shift—the graph shifts to the left if $\mathit{h}\mathbf{<}\mathbf{0}$ or right when $\mathit{h}\mathbf{>}\mathbf{0}$.

Example: Graph of $f\left(x\right)={\left(x-4\right)}^2 \left(gray\right), g\left(x\right)={\left(x+3\right)}^2 \left(green\right), and h\left(x\right)=-\frac{1}{2}{\left(x-1\right)}^2 \left(blue\right)$

4. $\mathit{y}\mathbf{=}\mathbf{\pm }\mathit{a}{\left(x-h\right)}^{\mathbf{2}}\mathbf{\pm }\mathit{k}\mathbf{,}\mathbf{ }\mathit{v}\mathit{e}\mathit{r}\mathit{t}\mathit{e}\mathit{x}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\left(h,k\right)$. The vertex form exactly indicates the transformation of the graph. The $\mathit{h}$ units indicate a horizontal shift while $\mathit{k}$ units indicate the vertical shift of the parabola. 

Example: Graph of $f\left(x\right)=\frac{1}{3}{\left(x+1\right)}^2+1 and g\left(x\right)=-\frac{1}{4}{\left(x-3\right)}^2-2$

Properties of Quadratic Function and its Graph

The properties of the quadratic function and its graph are as follows:

• The parabola opens upward if $\mathit{a}\mathbf{>}\mathbf{0}$. It opens downward if $\mathit{a}\mathbf{<}\mathbf{0}$.
• The vertex $\left(h,k\right)$ is the lowest/minimum point if the parabola opens upward.
• The vertex $\left(h,k\right)$ is the highest/maximum point if the parabola opens downward.
• The domain of quadratic functions is the set of all real numbers. 
• The range depends on the value of $\mathit{k}$ in the vertex $\left(h,k\right)$ and its direction of opening: $\left\{y|y\in ℝ, y\ge k\right\}$ if it opens upward, and $\left\{y|y\in ℝ, y\le k\right\}$ if it opens downward.
• The line or axis of symmetry is the line that separates the parabola into two halves or regions. 
• A parabola represented by a quadratic function has only one y-intercept.
• The zeros of the quadratic function can be solved by letting $y=0$. Common methods of solving for zeros are the same as solving for the roots of quadratic equations. These are factoring, extraction of roots, completing the square, using the Po-Shen Loh method, or using the quadratic formula.
• A parabola may have two x-intercepts (when there are two real distinct zeros), one x-intercept/tangent to the x-axis (when there is only one zero), or no x-intercept (when zeros are not real).
• The parabola gets narrower or closer to the line of symmetry when the absolute value of $\mathit{a}$ increases. It gets wider or moves away from the line of symmetry when the absolute value decreases.

Methods of Deriving the Equation of Quadratic Function

The quadratic function equation can be derived given the following information: table of values, the graph, some points on the graph, and the zeros. 

Graphing Quadratic Functions

Steps in sketching the graph of a quadratic function:

• Find the coordinates of the vertex by solving for the x-value using the formula $\mathit{x}\mathbf{=}\mathit{h}\mathbf{=}\mathbf{-}\frac{\mathbf{b}}{\mathbf{2}\mathbf{a}}$. Then, direct substitute the obtained x-value to the quadratic function and solve for the y-coordinate of the vertex. The formula $\mathit{k}\mathbf{=}\frac{\mathbf{4}\mathbf{a}\mathbf{c}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}}{\mathbf{4}\mathbf{a}}$ can also be used to find the y-coordinate.
• Construct a table of values using at least four points (consisting of two x-values greater than the x-coordinate of the vertex and two x-values less than the x-coordinate of the vertex).
• Sketch the graph by plotting the points and the vertex. Then, illustrate the graph by connecting the points, making a smooth curve.

Cheat Sheet

• Always inspect the sign of $\mathit{a}\mathbf{,}\mathbf{ }\mathit{h}\mathbf{,}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{k}$ in the equation of the quadratic function. These values tell the direction of the parabola's opening, shape, and translation.
• If $\left|a\right|\mathbf{<}\mathbf{1}$, the graph is wider than the graph of $\mathit{f}\left(x\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{.}$.
• If $\left|a\right|\mathbf{>}\mathbf{1}$, the graph is narrower than the graph of $\mathit{f}\left(x\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{.}$
• $\mathit{f}\left(x\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{8}}{\left(x-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{2}$ has a wider graph unlike the graph of $\mathit{g}\left(x\right)\mathbf{=}\mathbf{5}{\left(x-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{2}$.
• The parabola $\mathit{f}\left(x\right)\mathbf{=}\mathbf{2}{\left(x+1\right)}^{\mathbf{2}}\mathbf{-}\mathbf{1}$ has a vertex at $\left(-1,-1\right)$.
• To change the standard equation of the parabola $f\left(x\right)=a{x}^2+bx+c, where a\ne 0$, another option is Completing the Square. 
• Quadratic functions have the same properties as Absolute Value Functions except for the shape of their graphs.

Blunder Areas

• A quadratic function differs from a quadratic equation since the former involves independent and dependent variables while the latter consists of only the independent variable.
• A quadratic function whose graph does not touch the x-axis has no real zeros. 
• A quadratic function is an example of a not one-to-one function. Hence, its inverse is also not one-to-one.