Section 2.1 Intercepts; Symmetry; Graphing Key Equations


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1 Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the xaxis. 2. The xcoordinate of a point at which the graph crosses or touches the xaxis. 3. The xintercepts of the graph of an equation are those xvalues for which y = 0. y intercept is 1. The point where the line crosses (or intercepts) the yaxis. 2. The ycoordinate of a point at which the graph crosses or touches the yaxis. 3. The yintercepts of the graph of an equation are those yvalues for which x = 0. Notice that a point could be both the xintercept and yintercept. Finding xintercepts Step 1: Substitute 0 for y or. Step 2: Solve for the x variable. Finding yintercepts Step 1: Substitute 0 for x. Step 2: Solve for the y variable. Axis of symmetry is a line of symmetry for a graph If a graph has the xaxis symmetry, then. Step 1: Replace y by y in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the x axis if the equivalent equation results. If a graph has the yaxis symmetry, then. Step 1: Replace x by x in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the y axis if the equivalent equation results If a graph has the origin symmetry, then Step 1: Replace x by x and y by y in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the origin if the equivalent equation results CheonSig Lee Page 1
2 Exercises (Solution 2) The xintercepts of the graph of an equation are those xvalues for which y = 0. The yintercepts of the graph of an equation are those yvalues for which x = (Solution 3) For every point, ; Symmetric with respect to the yaxis is, Symmetric with respect to the xaxis is, Symmetric with respect to the origin is, 4. (Solution 4) 4 is an xintercept of this graph 4, 0 Symmetric with respect to the yaxis is 4, 0 Symmetric with respect to the xaxis is 4, 0 Symmetric with respect to the origin is 4, 0 CheonSig Lee Page 2
3 5. (Solution 5) 3, 4 If the graph is symmetric with respect to the yaxis, value changes, so 3, 4 If the graph is symmetric with respect to the xaxis, value changes, so 3, 4 If the graph is symmetric with respect to the origin, both value and value change, so 3, 4 6. (Solution 6) To find xintercepts of the graph of an equation, let y = 0 and solve for x. To find yintercepts of the graph of an equation, let x = 0 and solve for y. 7. (Solution 7) The xcoordinate of a point at which the graph crosses or touches the xaxis is an xintercept. The ycoordinate of a point at which the graph crosses or touches the yaxis is a yintercept. 8. (Solution 8) The statement is false because a graph of a circle is symmetric with respect to the xaxis, yaxis, and origin. CheonSig Lee Page 3
4 9. (Solution 9) xintercept yintercept Therefore, the intercepts are 2,0, 2,0, 0, 8 Step 1: Substitute 0 for y. Step 1: Substitute 0 for x. and the graph is shown as below when you plot the obtained all three points Step 2: Solve for x. Step 2: Solve for y So, yintercept is 0, So, xints are 2,0, 2,0 10. Plotting 3, 2 CheonSig Lee Page 4
5 11. (Solution 11) (a) (b) xintercepts are 3,0 and 3,0 yintercept is 0,3 Thus, intercepts are 3,0, 3,0, 0,3 Actually, we do not have enough information to decide the symmetry. We must assume that the graph has the symmetry. yaxis symmetry: Origin symmetry: CheonSig Lee Page 5
6 12. (Solution 12) Finding xintercept(s) Substitute 0 for y, then solve for x Let m be, then 0 and we have By acmethod, we have or Since m = x 2, m is a positive for all x. Thus, discard the proposed solution Therefore, the xintercepts are 4,0, 4,0 Finding yintercept(s) Substitute 0 for x, then solve for y Therefore, the yintercept is 0, 96 xintercept yintercept xintercept xaxis symmetry Replace y by y, then solve for y is different from the original equation, so the graph is not symmetric with respect to the xaxis yaxis symmetry Origin symmetry Replace x by x, then solve for y. Replace x by x and y by y, then solve for y The equivalent equation results, so the graph has the yaxis symmetry is different from the original equation, so the graph is not symmetric with respect to the origin. CheonSig Lee Page 6
7 13. (Solution 13) xintercept(s) Substitute 0 for y, then solve for x or Therefore, the xintercepts are 0,0, 7,0 and 7,0 Origin symmetry Replace x by x and y by y, then solve for y The equivalent equation results, so the graph has the origin symmetry. yintercept(s) Substitute 0 for x, then solve for y Thus, the yintercept is 0,0 Therefore, the intercepts are 0,0, 7,0, and 7,0 xaxis symmetry Replace y by y, then solve for y is different from the original equation, so the graph is NOT symmetric with respect to the xaxis. yaxis symmetry Replace x by x, then solve for y is different from the original equation, so the graph is NOT symmetric with respect to the yaxis. Graph the function. Step 1: Plot all intercepts obtained. Step 2: Find behavior of the end points. CheonSig Lee Page 7
8 Solving Quadratic Equations By Factoring By Square Root Property and Completing the square By Quadratic Formula Solving Quadratic Equations by Factoring Step 1: Factor out GCF, if possible Step 2: Determine the number of terms in the polynomial Step 3: Determine a factoring technique as follows: If the given polynomial is a Binomial, factoring by one of the following If the given polynomial is a Trinomial, factoring by acmethod If the given polynomial has 4 or more terms, factoring by Grouping Greatest Common Factor (GCF): The GCF is an expression of the highest degree that divides each term of the polynomial. The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial. Finding GCF Step 1: Find the prime factorization of all integers and integer coefficients Step 2: List all the factors that are common to all terms, including variables Step 3: Choose the smallest power for each factor that is common to all terms Step 4: Multiply these powers to find the GCF NOTE: If there is no common prime factor or variable, then the GCF is 1 (Example) Find the GCF for the following set of algebraic terms: {30x 4 y, 45x 3 y, 75x 5 y 2 } Therefore, GCF is CheonSig Lee Page 8
9 Factoring by acmethod is used for trinomials and some binomials Factoring acmethod will be discussed with examples Examples for positive ac 1. Factor the given polynomial Factor the given polynomial 5 4 Step 1: Factor out GCF. GCF = 1, so DONE! Step 1: Factor out GCF. GCF = 1, so DONE! 1, 7, 10 1, 5, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 4: 1 4, 2 2 Since ac is positive, there are two case; Since ac is positive, there are two case; Case 1: Both factors are positive Case 1: Both factors are positive Case 2: Both factors are negative Case 2: Both factors are negative Since b is positive, both factors are positive Since b is negative, both factors are negative 1 10,, 2 2 Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. Then numbers found in step 4 as coefficients. Then factoring by grouping. factoring by grouping Factor the given polynomial Factor the given polynomial Step 1: Factor out GCF. GCF = 2 Step 1: Factor out GCF. GCF = , 11, 5 2, 9, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 14: 1 14, 2 7 Since ac is positive, there are two case; Since ac is positive, there are two case; Case 1: Both factors are positive Case 1: Both factors are positive Case 2: Both factors are negative Case 2: Both factors are negative Since b is positive, both factors are positive Since b is negative, both factors are negative, , Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. numbers found in step 4 as coefficients. Then, factoring by grouping. Then, factoring by grouping CheonSig Lee Page 9
10 Examples for negative ac 1. Factor the given polynomial Factor the given polynomial 3 10 Step 1: Factor out GCF. GCF = 1, so DONE! Step 1: Factor out GCF. GCF = 1, so DONE! 1, 3, 10 1, 3, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 10: 1 10, 2 5 Since ac is negative, one factor is positive Since ac is negative, one factor is positive and the other is negative and the other is negative Since b is positive, bigger factor is positive Since b is negative, bigger factors is negative 1 10, 1 10, Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. Then numbers found in step 4 as coefficients. Then factoring by grouping. factoring by grouping Factor the given polynomial Factor the given polynomial Step 1: Factor out GCF. GCF = 2 Step 1: Factor out GCF. GCF = , 11, 5 2, 5, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 14: 1 14, 2 7 Since ac is negative, one factor is positive Since ac is negative, one factor is positive and the other is negative and the other is negative Since b is positive, bigger factor is positive Since b is negative, bigger factors is negative, , Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. numbers found in step 4 as coefficients. Then, factoring by grouping. Then, factoring by grouping Square Root Property If, then where a is a nonzero real number. Note: indicates that or CheonSig Lee Page 10
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