Analytic Geometry
Quadratic Equations and Functions
This document is intended to give an overview regarding the Analytic Geometry topic: Quadratic Equations and Functions. Students have dealt with two types of functions already. These functions are linear functions and exponential functions. We will compare quadratic functions to linear and exponential functions when we talk about rate of change. The Common Core Georgia Performance Standards (CCGPS) regarding quadratic functions are listed below.
STANDARDS ADDRESSED
Use complex numbers in polynomial identities and equations
MCC9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
Interpret the structure of expressions
MCC9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
Write expressions in equivalent forms to solve problems
MCC9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.★
MCC9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.★
Create equations that describe numbers or relationships
MCC9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential
functions.★
MCC9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
Solve equations and inequalities in one variable
MCC9-12.A.REI.4 Solve quadratic equations in one variable.
MCC9-12.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
MCC9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a andb.
Solve systems of equations
MCC9-12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Interpret functions that arise in applications in terms of the context
MCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★
MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified and exponential functions studied in Coordinate Algebra.)
Analyze functions using different representations
MCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.F.IF.7a Graph quadratic functions and show intercepts, maxima, and minima.
MCC9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context.
MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
Build a function that models a relationship between two quantities
MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
Build new functions from existing functions
MCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
Construct and compare linear, quadratic, and exponential models and solve problems
MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★
Summarize, represent, and interpret data on two categorical and quantitative variables
MCC9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.★
MCC9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.★
MCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★(Focus on quadratic functions; compare with linear and exponential functions studied in Coordinate Algebra.)
MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★
MCC9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (Focus on quadratic functions; compare with linear, and exponential models.)★
Georgia Virtual School also has a great tutorial regarding Quadratic Functions and Equations. It can be accessed using this link:
Graphing Quadratic Functions
Page 4 of Georgia Virtual School Site
When graphing quadratic equations, there are two main methods that are used.
1)One method is to create an input/output table and plot the points found. After plotting several points, a parabolic shape is created and we can draw a smooth curve through these points to graph the parabola.
2)The second method is to put the quadratic function in vertex form (if it isn’t already in the form) and plot it using transformations.
Here is a quadratic function plotted using both methods.
x / y /
-4 / -3
-3 / -6
-2 / -7
-1 / -6
0 / -3
1 / 2
2 / 9
3 / 18
/
To graph the functions using transformations, we have to consider the graph of y = x2 and how we can transform it to create the new function. To create the above function, the function y = x2 has been translated 7 units down and 2 units left to create the new function.
Video: A tutorial video on graphing using transformations can be found at
More Help: Page 4 of the Georgia Virtual School website on Analytic Geometry: Quadratic Functions provides a great overview of graphing quadratic functions.
Properties of Quadratic Functions
Page 5 of Georgia Virtual School Site
Previously, we discussed what the effects “a,” “h,” and “k” have on a quadratic function. This is for when the function is in vertex form. However, when a quadratic function is in standard form, y = ax2 + bx + c, it has to be handled differently. Of course, it is possible to convert a quadratic function from standard form to vertex form and that is a perfectly fine way of tackling the problem, but the following information pertains to quadratic functions in standard form.
y = ax2 + bx + c
The value “a”
The value “a” does exactly the same thing it did when then function is in vertex form. In fact, it has the exact same value also!
1)If “a” is positive, it tells us that the graph opens UPWARD (like a smiley face).
If “a” is negative, it tells us that the graph opens DOWNWARD (like a frowny face).
“a” is positive“a” is negative
The value “c”
The value “c” is simply the y-intercept.
Ex: y = 2x2 + 3x – 2
Looking at the function, the value of c
isc = -2. Looking at the graph, the graph
crosses the y-axis at y = -2.
The value of c will ALWAYS be the y-intercept.
Finding the Vertex
The vertex of a parabola when the function is in standard form can be found by first finding the axis of symmetry. The axis of symmetry is a vertical line that cuts the parabola in HALF. It will ALWAYS pass through the vertex.
The axis of symmetry has the equation .
Since we know the vertex lies on the axis of symmetry, and we now know the equation of the axis of symmetry, this means we know the x-value for the coordinate of the vertex! So, we now just substitute that value into the function to get the y-coordinate of the vertex.
Vertex: So, the x-coordinate is found by using . Substitute that value into the function to get the output. This is the y-coordinate.
For functions where “a” is positive, the vertex will be a minimum.
For functions where ”a” is negative, the vertex will be a maximum.
Ex: y = 2x2– 4x + 3
Axis of symmetry: . Substitute x = 1 into the function. y = 2(1)2 – 4(1) + 3 = 2(1) – 4(1) + 3 = 2 – 4 + 3 = 1.
So, the vertex of y = 2x2 – 4x + 3 is at the coordinate (1, 1) and is a minimum.
Domain and Range
The domain of a function is the set of allowable inputs for a function.
The range of a function is the set of outputs for a function.
If we consider the x-axis (the numbers from which we get our domain), it is simply a number line consisting of real numbers. *Using these numbers in a quadratic function will produce more real numbers.
*This is not always the case for other functions. For example, if we consider the function , we can not use negative numbers because when we take the square root of negative numbers we enter the complex number system.
So, the domain for ANY quadratic function will be -∞ ≤ x ≤ ∞, or All Reals, or (-∞ , ∞). These are three equivalent ways of stating the domain.
The range of a quadratic function is NOT the set of all real numbers. The reason is because of the vertex, or the absolute maximum/minimum.
If the quadratic function has a minimum, then the range will be .
If the quadratic function has a maximum, then the range will be
Remember, the value is simply the y-coordinate of the vertex.
Ex: State the domain and range of the function y = -3x2 + 12x – 4.
Domain: The domain will be All Reals because the function is quadratic.
Range: We need to find the vertex in order to state the range. So,
1)Find the Axis of symmetry
2)Substitute x = 2 into the function
y = -3(2)2 + 12(2) – 4 = -3(4) + 12(2) – 4 = -12 + 24 – 4 = 8.
The vertex is at (2 , 8) and it is a maximum because “a” is negative. So, the range is (-∞ , 8].
Interval of Increase and Decrease
If we think about a quadratic function as a roller coaster, half of the time it is going downhill (decreasing) and half of the time it is going uphill (increasing). The portion of the domain that the graph is going uphill is called the interval of increase. The portion of the domain that the graph is going downhill is called the interval of decrease.
Consider the function y = -3x2 + 12x – 4.
We know this graph is opened DOWNWARD because “a” is negative. So, the vertex is a maximum. So, to the left side of the vertex the graph will be INCREASING. To the right side of the vertex, the graph will be DECREASING.
Looking at the graph, we see that the vertex has an x-coordinate of 2. So, for the portion of the domain that is less than 2, the graph will be increasing. For the portion of the domain that is greater than 2, the graph will be increasing.
Interval of Increase: (-∞ , 2)
Interval of Decrease: (2, ∞).
End Behavior
When we talk about a functions end behavior, we are considering which direction the outputs are going as the inputs get large (more positive) or small (more negative). Because a quadratic function points in the same direction on each side of its graph, there are only two possibilities for the end behavior.
Either, as x increases, f(x) increasesand as x decreases, f(x) increases
or
as x increases, f(x) decreasesand as x decreases, f(x) decreases.
For this graph, as we look at the left side of the graph, we see that the graph is pointing upwards, so we say as x decreases, f(x) increases. If we look at the right side of the graph, we see it is also pointing upwards, so we say as x increase, f(x) increases. /
For this graph, as we look at the left side of the graph, we see that the graph is pointing downwards, so we say as x decreases, f(x) decreases. If we look at the right side of the graph, we see it is also pointing downwards, so we say as x increase, f(x) decreases.
This can simply be determined by knowing if the graph has a maximumor minimum. See the above graphs.
Rate of Change – Page 7 on Georgia Virtual School Site
Unlike linear functions, quadratic functions do not have a constant rate of change. However, we can approximate the rate of change between two points on a quadratic function by treating it like a linear function.
For any quadratic function, the rate of change between x1 and x2 can be calculated by using
Where x1 and x2 are inputs and f(x1) and f(x2) are the corresponding outputs.
Ex: Consider the function y = -3x2 + 12x – 4. Determine the rate of change between x = -1 to x = 3.
We need to determine f(-1) and f(3).
f(-1) = -3(-1)2 + 12(-1) – 4 = -3(1) + 12(-1) – 4 = -3 – 12 – 4 = -19.
f(3) = -3(3)2 + 12(3) – 4 = -3(9) + 12(3) – 4 = -27 + 36 – 4 = 5.
So, the rate of change from x = -1 to x = 3 is 6.
Determining Types of Solutions – Page 9 from Georgia Virtual School Site
The part of the quadratic formula that is under the radical is called the discriminant. Remembering what we know about radicals, we can determine the number and types of solutions that we will wind up with.
Discriminant: b2 – 4ac
1)If the discriminant is a positive perfect square: 2 real rational roots.
2)If the discriminant is a positive non-perfect square: 2 real irrational roots.
3)If the discriminant is zero: 1 real rational root.
4)If the discriminant is negative: 2 complex roots.
Example 1 / Example 2 / Example 3 / Example 4y = 3x2 – 12x + 4
b2–4ac = (-12)2–4(3)(4)
= 144 - 48
= 96
Positive, non-perfect square, so the function has 2 irrational roots. / y = -3x2 – 2x + 8
b2–4ac = (-2)2 - 4(-3)(8)
= 4 + 96
= 100
Positive, perfect square, so the function has 2 rational roots. / y = -4x2 + 5x – 9
b2–4ac = (5)2 – 4(-4)(-9)
= 25 – 144
= -119
Negative, so the function has 2 complex roots. / y = -4x2 + 8x -4
b2–4ac = (8)2 -4(-4)(-4)
= 64 – 64
= 0
Zero, so the function has 1 rational root.
Solving a Quadratic & Linear System
Page 6 of the Georgia Virtual School Site
The Georgia Virtual School Site covers how to solve a system of equations involving a linear function and a quadratic function using graphing. Just see page 6.
We will go over how to solve the system using algebra. Knowing how to solve a quadratic equation is absolutely essential in solving a system of equations involving a quadratic equation and linear equation. Let’s look at an example.
Ex. y = 2x2 + 8x + 2
y = -x – 2
By graphing, we can visually see where the two functions intersect. However, your answers may only be approximations if the intersections are not at “nice” places. Also, the accuracy depends on your attention to precision.
Algebra can eliminate these errors if careful attention to paid to make sure each step is executed correctly.
y = 2x2+ 8x + 2
y = -x – 2
Sometimes, the linear function is not expressed as one variable in terms of the other (one variable on a side by itself). So, it must be transformed so one variable is on a side by itself. This one is already done for us, so we will proceed.
Substitute the linear function into the quadratic function. (Replace the “y” in the quadratic function with what “y” equals in the linear function.)
-x – 2 = 2x2 + 8x + 2 Now make this equation equal to 0.
+x + 2 + x + 2
0 = 2x2 + 9x + 4Now, solve the equation.
0 = (2x + 1)(x + 4)
0 = 2x + 1 or 0 = x + 4
-1 = 2x -4 = x
-1/2 = x These solutions tell us the x-coordinates of the point(s) of intersection.
To find the y-coordinates, substitute the values into the linear function.
y = -(-1/2) – 2y = -(-4) – 2
y = 1/2 – 2y = 4 – 2
y = -1 ½y = 2These are the corresponding y-coordinates for the point(s) of
intersection.
The points of intersection of the line y = -x – 2 and the quadratic function y = 2x2 + 8x + 2 are (-1/2 , -1 ½) and (-4, 2).