9.1: GRAPHING QUADRATICS ALGEBRA 1
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1 9.1: GRAPHING QUADRATICS ALGEBRA 1
2 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like?
3 WHAT CAN I FIND FROM MY GRAPH? Equation for Axis of Symmetry In y = 2x 2 + 4x + 1, a = 2 and b = 4. Substitute these values into the equation of the axis of symmetry. x = b 2a x = 4 2(2) = 1 The axis of symmetry is x = 1. Coordinates of the Vertex Since the equation of the axis of symmetry is x = 1 and the vertex lies on the axis, the x coordinate of the vertex is 1. y = 2x 2 + 4x + 1 y = 2( 1) 2 + 4( 1) + 1 y = 2(1) y = 1 The vertex is at ( 1, 1). Simplify. Original equation Substitute. Y-Intercept: The point where the parabola crosses the y axis. This can be found by substituting a 0 in for x. Minimum: If lead coefficient is + (Parabola Opens Upward) then vertex is at BOTTOM. This is a minimum! Maximum: If lead coefficient is - (Parabola Opens Downward) then vertex is at TOP. This is a maximum! Domain: All x Values Range: All y Values Number of Solutions: This is based on # of times the Parabola CROSSES the x axis
4 HOW DO I GRAPH IN STANDARD FORM? y = ax 2 + bx + c 1. Find the x-coordinate of the vertex: x = b 2a (This is axis of symmetry a.o.s.) 2. Draw and fill out a table of values. Begin with the a.o.s. for x value! x a.o.s. y Shortcut - Find the y-intercept and plug in that ordered pair! 3. Plot 5 points and draw a smooth curve to connect them! ***USE ARROWS***
5 EXAMPLE 1: GRAPH y = x 2 + 4x 1 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
6 EXAMPLE 2: GRAPH y = 2x 2 + 8x 5 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
7 PRACTICE 1: GRAPH y = x 2 + 4x + 1 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
8 PRACTICE 2: GRAPH y = x 2 4x 8 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
9 HOW DO I GRAPH IN VERTEX FORM? y = a x h 2 + k 1. Find the vertex. Since the equation is in vertex form, the vertex will be at the point (Opposite of h, Same as k). 2. Draw and fill out a table of values. Begin with the a.o.s. for x value! x y Shortcut - Find the y-intercept and `plug in that ordered pair! 3. Plot 5 points and draw a smooth curve to connect them! ***USE ARROWS***
10 EXAMPLE 3: GRAPH y = 2(x + 1) What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
11 EXAMPLE 4: GRAPH y = 3(x 2) 2 4 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
12 PRACTICE 3: GRAPH y = (x + 5) 2 2 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
13 PRACTICE 4: GRAPH y = (x 3) 2 +1 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce?
14 HOW DO I GRAPH IN INTERCEPT FORM? y = a(x p)(x q) 1. Identify intercepts: p, 0 & (q, 0) 2. Find the x-coordinate of the vertex: x = p+q 3. Find the y-coordinate: Plug in x. 4. Graph the line of symmetry: x = # 5. Plot 2 more points and draw curve 2
15 EXAMPLE 5: GRAPH y = (x 2)(x + 4)
16 EXAMPLE 6: GRAPH y = (x 3)(x + 2)
17 PRACTICE 5: GRAPH y = (x 1)(x + 3)
18 PRACTICE 6: GRAPH y = (x + 3)(x + 5)
19 HOMEWORK 9.1 Packet Page DUE MONDAY, FEBRUARY 23 rd!
20 9.2 SOLVING QUADRATIC EQUATIONS BY GRAPHING Quadratic Equation an equation of the form ax 2 + bx + c = 0, where a 0 Solve by Graphing The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by graphing the related quadratic function f(x) = ax2 + bx + c and finding the x-intercepts or zeros of the function.
21 HOW TO SOLVE BY GRAPHING Example 1: Solve x 2 + 4x + 3 = 0 by graphing. Graph the related function f(x) = x 2 + 4x + 3. ***The Equation is in STANDARD FORM, so use a TABLE OF VALUES to Graph! To solve x 2 + 4x + 3 = 0, you need to know where f(x) = 0. This occurs at the x-intercepts, 3 and 1. The solutions are 3 and 1.
22 EXAMPLE 1: SOLVE BY GRAPHING x 2 x 12 = 0 X Y ( ) 2 ( ) 12 = 0 ( ) 2 ( ) 12 = 0 ( ) 2 ( ) 12 = 0 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce? What are the solutions?
23 EXAMPLE 2: SOLVE BY GRAPHING x 2 + 7x + 12 = 0 X Y ( ) 2 + 7( ) + 12 = 0 ( ) 2 + 7( ) + 12 = 0 ( )² + 7( ) + 12 = 0 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce? What are the solutions?
24 PRACTICE 1: SOLVE BY GRAPHING x 2 4x + 5 = 0 X Y ( ) 2 4( ) + 5 = 0 ( ) 2 4( ) + 5 = 0 ( ) 2 4( ) + 5 = 0 What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce? What are the solutions?
25 ON YOUR OWN: SOLVE BY GRAPHING x 2 2x + 3 = 0 X Y What is the vertex? What is the equation for the axis of symmetry? Is the vertex a maximum or a minimum? What are the Domain and Range? How many solutions does the equation produce? What are the solutions?
26 9.3 TRANSFORMATION OF QUADRATIC FUNCTIONS Translations: A translation is a change in the position of a figure either up, down, left, right, or diagonal. Adding or subtracting constants in the equations of functions translates the graphs of the functions. The graph of g(x) = (x h) 2 is the graph of f(x) = x 2 translated horizontally. If h > 0, the graph of f(x) = x 2 is translated h units to the right. If h < 0, the graph of f(x) = x 2 is translated h units to the left. The graph of g(x) = x 2 + k translates the graph of f(x) = x 2 vertically. If k > 0, the graph of f(x) = x 2 is translated k units up. If k < 0, the graph of f(x) = x 2 is translated k units down.
27 TRANSFORMATIONS NAME DATE PERIOD Example: Describe how the graph of each function is related to the graph of f(x) = x 2.le a. g(x) = x The value of k is 4, and 4 > 0. Therefore, the graph of g(x) = x is a translation of the graph of f(x) = x 2 up 4 units b. g(x) = (x + 3) 2 The value of h is 3, and 3 < 0. Thus, the graph of g(x) = (x + 3) 2 is a translation of the graph of f(x) = x 2 to the left 3 units.
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