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 x-axis. 2. The x-coordinate of a point at which the graph crosses or touches the x-axis. 3. The x-intercepts of the graph of an equation are those x-values for which y = 0. y intercept is 1. The point where the line crosses (or intercepts) the y-axis. 2. The y-coordinate of a point at which the graph crosses or touches the y-axis. 3. The y-intercepts of the graph of an equation are those y-values for which x = 0. Notice that a point could be both the x-intercept and y-intercept. Finding x-intercepts Step 1: Substitute 0 for y or. Step 2: Solve for the x variable. Finding y-intercepts 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 x-axis 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 y-axis 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 Cheon-Sig Lee Page 1

2 Exercises (Solution 2) The x-intercepts of the graph of an equation are those x-values for which y = 0. The y-intercepts of the graph of an equation are those y-values for which x = (Solution 3) For every point, ; Symmetric with respect to the y-axis is, Symmetric with respect to the x-axis is, Symmetric with respect to the origin is, 4. (Solution 4) 4 is an x-intercept of this graph 4, 0 Symmetric with respect to the y-axis is 4, 0 Symmetric with respect to the x-axis is 4, 0 Symmetric with respect to the origin is 4, 0 Cheon-Sig Lee Page 2

3 5. (Solution 5) 3, 4 If the graph is symmetric with respect to the y-axis, -value changes, so 3, 4 If the graph is symmetric with respect to the x-axis, -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 x-intercepts of the graph of an equation, let y = 0 and solve for x. To find y-intercepts of the graph of an equation, let x = 0 and solve for y. 7. (Solution 7) The x-coordinate of a point at which the graph crosses or touches the x-axis is an x-intercept. The y-coordinate of a point at which the graph crosses or touches the y-axis is a y-intercept. 8. (Solution 8) The statement is false because a graph of a circle is symmetric with respect to the x-axis, y-axis, and origin. Cheon-Sig Lee Page 3

4 9. (Solution 9) x-intercept y-intercept 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, y-intercept is 0, So, x-ints are 2,0, 2,0 10. Plotting 3, 2 Cheon-Sig Lee Page 4

5 11. (Solution 11) (a) (b) x-intercepts are 3,0 and 3,0 y-intercept 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. y-axis symmetry: Origin symmetry: Cheon-Sig Lee Page 5

6 12. (Solution 12) Finding x-intercept(s) Substitute 0 for y, then solve for x Let m be, then 0 and we have By ac-method, we have or Since m = x 2, m is a positive for all x. Thus, discard the proposed solution Therefore, the x-intercepts are 4,0, 4,0 Finding y-intercept(s) Substitute 0 for x, then solve for y Therefore, the y-intercept is 0, 96 x-intercept y-intercept x-intercept x-axis 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 x-axis y-axis 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 y-axis symmetry is different from the original equation, so the graph is not symmetric with respect to the origin. Cheon-Sig Lee Page 6

7 13. (Solution 13) x-intercept(s) Substitute 0 for y, then solve for x or Therefore, the x-intercepts 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. y-intercept(s) Substitute 0 for x, then solve for y Thus, the y-intercept is 0,0 Therefore, the intercepts are 0,0, 7,0, and 7,0 x-axis 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 x-axis. y-axis 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 y-axis. Graph the function. Step 1: Plot all intercepts obtained. Step 2: Find behavior of the end points. Cheon-Sig 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 ac-method 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 Cheon-Sig Lee Page 8

9 Factoring by ac-method is used for trinomials and some binomials Factoring ac-method 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 Cheon-Sig 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 Cheon-Sig Lee Page 10

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