Objectives and Standards
1. We will identify key features of a quadratic graph and sketch a graph based on the key features. MAFS.912.F-IF.2.4
2. We will determine if the function is quadratic based on a table, intercepts, and a vertex. MAFS.912.F-IF.3.8
3. We will graph a quadratic equation using vertex form and other key features. MAFS.912.F-IF.3.7, MAFS.912.A.REI.2.4
4. We will determine how quadratic functions transform on the dependent and independent variables. MAFS.912.F-BF.2.3
5. We will determine the zeros of a polynomial function. MAFS.912.A-APR.2.3
6. We will determine the end behavior of a polynomial function. MAFS.912.F-IF.3.7.c
7. We will determine how to graph polynomial functions. MAFS.912.F-IF.3.7
MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MAFS.912.F-IF.2.4 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.
MAFS.912.F-IF.3.7.a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima. This section focuses on quadratic functions.
MAFS.912.F-IF.3.8.a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
MAFS.912.A.REI.2.4 Solve quadratic equations with one variable.
a. 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.
b. 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 and b.
MAFS.912.A-REI.4.11 Explain why the x−coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.
Topics
Key Features of a Quadratic FunctionWhen determining the key features of a quadratic function, using a graph will be the most helpful.
|
Determine if the Function is a Quadratic Based on a Table, Vertex, and InterceptsWhen creating a graph of a quadratic function, you must be able to use the key features.
If given an equation written in standard form, you must have your key features from that equation.
www.virtualnerd.com/algebra-1/quadratic-equations-functions/table-example.php |
Graph a Quadratic Function using Vertex FormIf given a quadratic function in vertex form:
f(x) = a(x-h)^2 + k
|
Quadratic Function TransformationsTransformations on quadratic functions occur on the independent and dependent variables. Independent transformations happen on the x axis and dependent transformations happen on the y axis.
Because 𝒇(𝒙) represents the dependent variable and in each function we are performing an operation on 𝒇 𝒙 . 𝑔(𝑥)= 𝑓(𝑥)+ 2 vertical shift up two units ℎ (x)=f(x) - 2 vertical shift down two units 𝑚(𝑥)= 2𝑓(𝑥) vertical compression 𝑛(𝑥)=1/2𝑓(𝑥) vertical stretch 𝑝(𝑥)= −𝑓(𝑥) reflection over the x-axis
𝑔(𝑥)= 𝑓(𝑥 + 2) horizontal shift left two units ℎ(𝑥)= 𝑓(𝑥 − 2) horizontal shift right two units 𝑚(𝑥)= 𝑓(2𝑥) horizontal compression 𝑛(𝑥)=𝑓(1/2x) horizontal stretch |
Zeros of a Polynomial FunctionIn order to find the zeros of a polynomial function, you must find the x-intercepts. On a graph, that is where the polynomial function crosses the x-axis. If you're given an equation in standard form, you need to factor and solve for x. If you're equation is already factored into binomials, set each binomial equal to 0 and solve for x. If you're given the domain of a function and need to solve for the range, substitute the domain values into the function and solve. Simplify using order of operations.
|
End Behavior of a Polynomial Function |
Graphing a Polynomial Function |
End behavior of a function is determined by the exponent and if the function is positive or negative.
If the function is positive and has an even exponent, the ends of the function will both be facing up. If the function is negative and has an even exponent, the ends of the function will both be facing down. If the function is positive and has an odd exponent, then the function starts down and ends up. If the function is negative and has an odd exponent, then the function starts up and ends down. |
The last objective focused on graphing polynomial functions. This objective was embedded with the previous two objectives on polynomial functions. We are generating general sketches for these polynomials functions. Once you find the zeros and determine your end behavior, graph the points and draw a wavy function that connects the zeros with the correct end behavior.
|
Reflection
What I saw the students struggle with the most was transforming quadratic equations into vertex form. This was he most difficult because it had many parts to the process. Another topic that was difficult was determining end behavior of a polynomial function when given how x values were increasing and decreasing. I found that the students also really liked studying about Polynomial functions since it stemmed from graphing quadratic functions.