Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1
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1 Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the next three lessons. Any function where the highest exponent is a 2 is called a function. The shape of the graph of a quadratic function is called a A parabola has vertical. That means that there is a vertical line that can be drawn so that both sides of the parabola perfectly match up. That line is called the ALWAYS x =. It is Every quadratic function has a turning point. This is called the. Each vertex of a quadratic function is either a or. This point MUST lie on the axis of symmetry. In fact, the x-value of the vertex IS the equation for the axis of symmetry. The point(s) on graph where the parabola touches/intersects the x-axis are called or or or A quadratic function may have,, or x-intercepts. The point on the graph where the graph of a parabola intersects the y- axis is called the. There can only be y-intercept.
2 Example #1 a) How many x-intercepts does the graph have? What are they? b) What is the y-intercept? c) Draw the axis of symmetry. What is the equation for the axis of symmetry? d) What are the coordinates of e) Is the vertex a maximum the vertex of the parabola? or a minimum? f) Here is a table of values for the given function. Fill in the missing values. x 1 f(x)
3 Example #2 a) How many x-intercepts does the graph have? What are they? b) What is the y-intercept? c) Draw the axis of symmetry. What is the equation for the axis of symmetry? d) What are the coordinates of e) Is the vertex a maximum the vertex of the parabola? or a minimum? f) Here is a table of values for the given function. Fill in the missing values. x f(x)
4 Compare the graphs from Example #1 and #2. What is the major difference between the two graphs? The equation for graph #1 is f(x) = x 2 4x + 3. The equation for graph #2 is f(x) = -x 2 4x. You can determine if a graph is going to open upward or downward based on the sign that comes on the number in front of the squared term. This is called the. If the leading coefficient is, the graph will open upward. If the graph opens upward, the vertex will be at the bottom of the parabola and therefore will be a If the leading coefficient is, the graph will open downward. If the graph opens downward, the vertex will be at the top of the parabola and therefore will be a
5 Day #2 A parabola has vertical. That means that there is a vertical line that can be drawn so that both sides of the parabola perfectly match up. That line is called the. It is ALWAYS x = Every quadratic function has a turning point. This is called the. Each vertex of a quadratic function is either a or. This point MUST lie on the axis of symmetry. In fact, the x-value of the vertex IS the equation for the axis of symmetry. How to find the axis of symmetry from the equation To find the axis of symmetry, use the formula x = b #1 Find the axis of symmetry of f(x) = x x 17 2a 2 Find the axis of symmetry of f(x) = x 2 8x + 19 #3 Find the axis of symmetry of f(x) = -x 2 7x + 19
6 Try these #4 Find the axis of symmetry of f(x) = x x 11.5 #5 Find the axis of symmetry of f(x) = -2x x #6 Find the axis of symmetry of f(x) = -4x x #7 Find the axis of symmetry of f(x) = 5x 2 40x + 18 #8 Find the axis of symmetry of f(x) = -x x + 12 #9 Find the axis of symmetry of f(x) = -3x x 3
7 How to find the axis of symmetry from the x-intercepts The axis of symmetry is located EXACTLY between the two x-intercepts. If you know the x- intercepts/roots/zeros/solutions to a quadratic equation and want to know the axis of symmetry, simply the two numbers together and divide by 2! #1 If the x-intercepts of a quadratic equation are located at 8 and 2, what is the equation for the axis of symmetry? #2 If the solutions of a quadratic equation are located at -12 and 4, what is the equation for the axis of symmetry? #3 What is the equation for the axis of symmetry for the equation f(x) = x 2 6x 16? #4 If the zeros of a quadratic function, F, are -3 and 5, what is the equation of the axis of symmetry of F? Justify your answer. [Jan 2018 #29] #5 If the x-intercepts of a quadratic equation are located at -11 and 3, what is the equation for the axis of symmetry? #6 If the solutions of a quadratic equation are located at 8 and 1, what is the equation for the axis of symmetry?
![Day #3 Looking at graphs of quadratic functions #1 On the set of axes below, draw the graph of y = x 2 4x 1. [June 2016 #27] [2 points] State the equation of the axis of symmetry.](/docs-images/88/117183177/images/8-1.jpg "#2 The graph representing a function is shown below. Which function has a minimum that is less than the one shown in the graph?")
8 Day #3 Looking at graphs of quadratic functions #1 On the set of axes below, draw the graph of y = x 2 4x 1. [June 2016 #27] [2 points] State the equation of the axis of symmetry. #2 The graph representing a function is shown below. Which function has a minimum that is less than the one shown in the graph? (1) y = x 2 6x + 7 (2) y = x (3) y = x 2 2x 10 (4) y = x 8 + 2
9 #3 Graph f(x) = x and g(x) = -x on the grid below. Does f(-2) = g(-2)? Use your graph to explain why or why not. [Jan 2017 #33]
10 #4 Graph the function f(x) = -x 2 6x on the set of axes below. [June 2017 #26] State the coordinates of the vertex of the graph.
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so f can now be rewritten as a product of g(x) = x 2 and the previous piecewisedefined
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It is than the graph of y= x if a > 1.
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