Week 2 Handout MAC 1140 Professor Niraj Wagh J Section 3.2 Quadratic Functions & Their Graphs Quadratic Function: Standard Form 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 numbers and a 0. To find the vertex (the highest or lowest point of a parabola) of standard form we first find the x coordinate (axis of symmetry) by computing x = b. Then 2a after you find the x coordinate you plug it back into the function to solve for y. Essentially # Vertex = b 2a, f # b& & % % (( $ $ 2a' ' The Y-coordinate is the actual MAX or MIN of the function. N. Wagh 1
Quadratic Function: Vertex Form f(x) = a(x h) 2 + k Vertex: (h, k) Axis of Symmetry: x = h The k value is the max or min of the parabola Conversion Between Forms Convert Standard Form -> Vertex Form: Complete the Square. Convert Vertex Form -> Standard Form: Distribute & Simplify. Completing the Square Steps Step 1. Make sure a = 1. If it does not = 1 then divide both sides of the equation by the value of a to make it 1. Step 2. Isolate all variable terms on one side of the equation, everything else on the other side! Step 3. Take half off the b term (this is the term with one x) and square it. Step 4. Add this number to BOTH sides of the equation. Step 5. Factor the resulting PERFECT square trinomial. Step 6. Use the Square Root Property to solve for x. Let s see a few examples of this N. Wagh 2
For the quadratic function below, (a) Write the function in vertex form (convert from standard to vertex). (b) Find the axis of symmetry and vertex of the parabola. (c) Graph the function. (d) Based on the graph, find the domain and range. (e) Based on the graph, determine the interval of increase and decrease. (f) State whether the vertex if a max or min point and give the corresponding max or min value of the function. 1. f(x) = 2x * 2x 24 N. Wagh 3
Find the equation of the quadratic function satisfying the given conditions. Express your answer in standard form, however it may be helpful to start out with vertex form first. 2. Vertex (-1, -4); through the point (5, 104) Circles A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle. Forms The standard form of an equation of a circle with radius r and center (h, k) is (x h) 2 + (y k) 2 = r 2 The general form of the equation of a circle is x 2 + y 2 + ax + by + c = 0 N. Wagh 4
To go from general form to standard form we have to complete the square: 1. Make sure that the coefficients for x 2 and y 2 are 1. 2. Rearrange so that you get the form: (x 2 + ax)+ (y 2 + by) = c 3. Take half of the a and b term, these are the coefficients of the x and y terms respectively. 4. Square those numbers and add it each respective number within the parentheses. 5. Now add the numbers to the other side of the equation. 6. Factor each and simplify. Let s see a few examples of this 3. Find the standard form (center-radius) of the equation of the circle with the given center and radius. Then graph the circle. Center: (1, -2), Radius: 4 4. Find the standard form (center-radius) of the equation of the circle. Determine the center, radius, and graph the circle. (a) x * + y * + 2x + 2y 23 = 0 N. Wagh 5
Now try it! (b) x * + y * + 6x 6y + 2 = 0 Wonderful! We ve completed the examples for this section! Now work on HW 3.2 in MyMathLab. If you have any questions, please let me know! I will be more than happy to help you! My student engagement hours are listed in the syllabus. J N. Wagh 6
Section 3.3 Quadratic Equations & Inequalities Zero-Product Property If ab = 0, then a = 0 or b = 0. è Whenever you are asked to find the ZEROS, X-INTERCEPTS, or SOLUTIONS, you can use the zero-product property. Square-Root Property If b is a real number and if a 2 = b, then a = ± b. The Quadratic Formula A quadratic equation written in the form ax 2 + bx + c = 0 has the solutions: x = b ± b2 4ac 2a X EQUALS NEGATIVE B PLUS OR MINUS THE SQUARE ROOT OF B SQUARED MINUS FOUR TIMES A TIMES C ALL OVER 2 TIMES A! The quadratic formula tells you what the solutions of a quadratic equation are. How and when to use? The quadratic formula is used when you cannot factor. Or perhaps you re not sure how to factor the quadratic equation then you can use the quadratic formula. In order to use it, the quadratic equation MUST be in standard form (ax 2 + bx + c). N. Wagh 7
The Discriminant The discriminant of a quadratic equation is defined as b 2-4ac. It tells you the number of real solutions the quadratic equation has. If the discriminant is greater than zero, è The quadratic function has two distinct real solutions. If the discriminant is less than zero, è The quadratic function has no real solutions (you may have complex). If the discriminant is equals zero, è The quadratic function has one real solution. Quadratic Inequalities A quadratic inequality is an inequality that can be written in the form: ax 2 + bx + c < 0 OR ax 2 + bx + c 0 OR ax 2 + bx + c > 0 OR ax 2 + bx + c 0 Steps for Solving Quadratic Inequalities 1. Move all the terms onto one side. 2. Factor if needed. 3. Find the zeros. 4. Create a sign chart. 5. Determine which intervals satisfy the original inequality. N. Wagh 8
3.3 Examples Solve each equation. You will need to decide which is the best property to solve. 1. x * = 16 2. 3x * 2x = 0 3. (5x 3) * = 3 4. x(12x + 11) = 2 5. 3 4 x* 3 4 x = 24 6. x* + 8x + 13 = 0 Solve by completing the square (review notes from previous class). 7. 2x * + 6x 3 = 0 N. Wagh 9
Find the discriminant. Use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not solve the equation. 8. 4x * = 6x + 3 Solve the following quadratic inequalities. 9. x * + 4x + 3 0 N. Wagh 10
10. 2x * x + 3 < 0 Wonderful! We ve completed the examples for this section! Now work on HW 3.3 in MyMathLab. If you have any questions, please let me know! I will be more than happy to help you! My student engagement hours are listed in the syllabus. J N. Wagh 11