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Quadratic Functions and Their Graphs

Identify the intercepts, vertex, and axis of symmetry

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Quadratic Functions and Their Graphs

Suppose \begin{align*}p(m)=\frac {1}{12}m^2 - m + 5\end{align*} represents the price of a gallon of gas as a function of the month of the year. Are you able to graph this function? What shape does the graph of the function make? What are the domain and range of the function? What is the vertex of the graph? 

Graphs of Quadratic Functions

Previous Concepts introduced the concept of factoring quadratic trinomials of the form \begin{align*}0=ax^2+bx+c\end{align*}. This is also called the standard form for a quadratic equation. The most basic quadratic equation is \begin{align*}y=x^2\end{align*}. The word quadratic comes from the Latin word quadrare, meaning “to square.” By creating a table of values and graphing the ordered pairs, you find that a quadratic equation makes a \begin{align*}U\end{align*}-shaped figure called a parabola. 

The most basic quadratic equation is \begin{align*}y=x^2\end{align*} and can be graphed using the following table of its values:

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
–2 4
–1 1
0 0
1 1
2 4

By graphing the points in the table, you can see that the shape is approximately like the graph below. This shape is called a parabola.

      

 

The Anatomy 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. All parabolas have a vertex, the ordered pair that represents the bottom (or the top) of the curve.

The vertex of a parabola is the ordered pair \begin{align*}(h, k)\end{align*}.

Because the line of symmetry is a vertical line, its equation has the form \begin{align*}x=h\end{align*}, where \begin{align*}h=\end{align*} the \begin{align*}x-\end{align*}coordinate of the vertex.

As with linear equations, the \begin{align*}x\end{align*}-intercepts of a quadratic function are where the graph intersects the \begin{align*}x\end{align*}-axis. The \begin{align*}y\end{align*}-value is zero at the \begin{align*}x\end{align*}-intercepts.

An equation of the form \begin{align*}y=ax^2\end{align*} forms a parabola.

If \begin{align*}a\end{align*} is positive, the parabola will open upward. The vertex will be a minimum.

If \begin{align*}a\end{align*} is negative, the parabola will open downward. The vertex will be a maximum.

The variable \begin{align*}a\end{align*} in the equation above is called the leading coefficient of the quadratic equation. Not only will it tell you if the parabola opens up or down, but it will also tell you the width.

If \begin{align*}a>1\end{align*} or \begin{align*}a<-1\end{align*}, the parabola will be narrow about the line of symmetry.

If \begin{align*}-1< a < 1\end{align*}, the parabola will be wide about the line of symmetry.

Let's find the \begin{align*}x\end{align*}-intercepts of the quadratic function \begin{align*}y=x^2+5x-6\end{align*}:

To find the \begin{align*}x\end{align*}-intercepts, set \begin{align*}y\end{align*} equal to zero and solve for \begin{align*}x\end{align*} by factoring.

\begin{align*}0=x^2+5x-6=x^2+-1x+6x-6=x(x-1)+6(x-1)=(x+6)(x-1)\end{align*}

This means that

\begin{align*}0=x+6 \Rightarrow x=-6\end{align*} or \begin{align*}0=x-1 \Rightarrow x=1\end{align*}.

Thus the \begin{align*}x\end{align*}-intercepts are -6 and 1.

The Vertex of a Quadratic Equation in Standard Form

The \begin{align*}x-\end{align*}coordinate of the vertex of \begin{align*}0=ax^2+bx+c\end{align*} is \begin{align*}x=-\frac{b}{2a}\end{align*}.

Let's determine the direction, shape and vertex of the parabola formed by \begin{align*}y=-\frac{1}{2} x^2\end{align*}.

The value of \begin{align*}a\end{align*} in the quadratic equation is \begin{align*}-\frac{1}{2}\end{align*}.

  • Because \begin{align*}a\end{align*} is negative, the parabola opens downward.
  • Because \begin{align*}a\end{align*} is between –1 and 1, the parabola is wide about its line of symmetry.
  • Because there is no \begin{align*}b\end{align*} term, \begin{align*}b=0\end{align*}. Substituting this into the equation for the \begin{align*}x\end{align*}-coordinate of the vertex, \begin{align*}x=-\frac{b}{2a}=-\frac{0}{2a}=0\end{align*}. (Note: It does not matter what \begin{align*}a\end{align*} equals; since \begin{align*}b=0\end{align*}, the fraction equals zero.) To find the \begin{align*}y\end{align*}-coordinate, substitute the \begin{align*}x\end{align*}-coordinate into the equation:

\begin{align*}y=-\frac{1}{2} x^2=-\frac{1}{2}\cdot 0^2=0\end{align*}

The vertex is \begin{align*}(0,0)\end{align*}.

    

 

Domain and Range

Several times throughout this textbook, you have experienced the terms domain and range. Remember:

  • Domain is the set of all inputs (\begin{align*}x-\end{align*}coordinates).
  • Range is the set of all outputs (\begin{align*}y-\end{align*}coordinates).

The domain of every quadratic equation is all real numbers \begin{align*}(\mathbb{R})\end{align*}. The range of a parabola depends upon whether the parabola opens up or down.

If \begin{align*}a\end{align*} is positive, the range will be \begin{align*}y \ge k\end{align*}.

If \begin{align*}a\end{align*} is negative, the range will be \begin{align*}y \le k\end{align*}, where \begin{align*}k= \text{the }y-\end{align*}coordinate of the vertex.

Let's find the range of the quadratic function \begin{align*}y=-2x^2+16x+5\end{align*}.

To find the range, we must find the \begin{align*}y\end{align*}-value of the vertex. Using the formula given above, we can find the \begin{align*}x\end{align*}-value of the vertex, and use that to find the \begin{align*}y\end{align*}-value of the vertex.

Since the \begin{align*}x\end{align*}-value of the vertex is \begin{align*}x=-\frac{b}{2a}\end{align*}, we get \begin{align*}x=-\frac{16}{2\cdot -2}=4.\end{align*}

Now, substitute 4 into the function:

\begin{align*}y=-2x^2+16x+5=-2(4)^2+6(4)+5=-2(16)+64+5=-32+69=37\end{align*}

The \begin{align*}y\end{align*}-value of the vertex is 37, and since \begin{align*}a=-2\end{align*}, the parabola is facing down, so 37 is the highest possible value for the range.

Thus, the range is \begin{align*}y\le 37.\end{align*}

Examples

Example 1

Earlier, you were told that \begin{align*}p(m)=\frac {1}{12}m^2 - m + 5\end{align*} represents the price of a gallon of gas as a function of the month of the year. Are you able to graph this function? What shape does the graph of the function make? What are the domain and range of the function? What is the vertex of the graph?

This is a quadratic function and the graph of the function is a parabola. This function can be graphed using a table of values and information about the direction it opens and its vertex. 

Since \begin{align*}a>0\end{align*}, this parabola opens upward. 

Also, because \begin{align*}a\end{align*} is between -1 and 1, the parabola is wide about its line of symmetry.

 

The \begin{align*}x-\end{align*}coordinate of the vertex can be found with the formula \begin{align*}x=-\frac{b}{2a}\end{align*}

\begin{align*}\frac{-(-1)}{2(\frac{1}{12})}=\frac{1}{\frac{1}{6}}= 1\times \frac{6}{1}= 6\end{align*}

To find the \begin{align*}y-\end{align*}coordinate of the vertex, plug in 6 to the function.

\begin{align*}p(m)&=\frac {1}{12}m^2 - m + 5\\ p(6)&=\frac {1}{12}(6)^2 - (6) + 5\\ p(6)&=\frac {1}{12}(36) -6 + 5\\ p(6)&=3 -1\\ p(6)&=2 \end{align*}

The vertex of this function is (-6, 2). 

The domain of this function is all real numbers because any quadratic has the range of all real numbers. The range of this function is \begin{align*}y \ge 2\end{align*} because the graph opens upward and the \begin{align*}y-\end{align*}value of the vertex is 2.

License: CC BY-NC 3.0

Example 2

Determine the direction, vertex and range of \begin{align*}y=7x^2+14x-9\end{align*}.

Since \begin{align*}a=7\end{align*} is positive, the direction of the parabola is upward. Now we find the vertex:

\begin{align*}x=-\frac{b}{2a}=-\frac{14}{2\cdot 7}=-\frac{14}{14}=-1.\end{align*}

Now, substitute \begin{align*}x=-1\end{align*} into the quadratic function:

\begin{align*}y=7x^2+14x-9=7(-1)^2+14(-1)-9=7(1)-14-9=7-23=-16.\end{align*}

Thus, the vertex is (-1, -16).

Since the parabola faces up, and the \begin{align*}y\end{align*}-value of the vertex is -16, the range is \begin{align*}y\ge -16\end{align*}.

Review

  1. Define the following terms in your own words.
    1. Vertex
    2. Line of symmetry
    3. Parabola
    4. Minimum
    5. Maximum
  1. Without graphing, how can you tell if \begin{align*}y=ax^2+bx+c\end{align*} opens up or down?

Graph the following equations by making a table. Let \begin{align*}-3 \le x \le 3\end{align*}. Determine the range of each equation.

  1. \begin{align*}y=2x^2\end{align*}
  2. \begin{align*}y=-x^2\end{align*}
  3. \begin{align*}y=x^2-2x+3\end{align*}
  4. \begin{align*}y=2x^2+4x+1\end{align*}
  5. \begin{align*}y=-x^2+3\end{align*}
  6. \begin{align*}y=x^2-8x+3\end{align*}
  7. \begin{align*}y=x^2-4\end{align*}

Does the graph of the parabola open up or down?

  1. \begin{align*}y=-2x^2-2x-3\end{align*}
  2. \begin{align*}y=3x^2\end{align*}
  3. \begin{align*}y=16-4x^2\end{align*}

Find the \begin{align*}x-\end{align*}coordinate of the vertex of the following equations.

  1. \begin{align*}x^2-14x+45=0\end{align*}
  2. \begin{align*}8x^2-16x-42=0\end{align*}
  3. \begin{align*}4x^2+16x+12=0\end{align*}
  4. \begin{align*}x^2+2x-15=0\end{align*}

Graph the following functions by making a table of values. Use the vertex and \begin{align*}x-\end{align*}intercepts to help you pick values for the table.

  1. \begin{align*}y=4x^2-4\end{align*}
  2. \begin{align*}y=-x^2+x+12\end{align*}
  3. \begin{align*}y=2x^2+10x+8\end{align*}
  4. \begin{align*}y=\frac{1}{2} x^2-2x\end{align*}
  5. \begin{align*}y=x-2x^2\end{align*}
  6. \begin{align*}y=4x^2-8x+4\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.1. 

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Vocabulary

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.

leading coefficient of a parabola

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".

quadratic function

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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