1 / 27

Algebra 1B Chapter 9

Learn how to solve quadratic equations by graphing the related function, finding the vertex and zeros, and determining the real-number solutions. Examples provided.

christinad
Download Presentation

Algebra 1B Chapter 9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Algebra 1BChapter 9 Solving Quadratic Equations By Graphing

  2. Warm Up 1. Graph y = x2 + 4x + 3. 2. Identify the vertex and zeros of the function above. vertex:(–2 , –1); zeros:–3, –1

  3. Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. When writing a quadratic function as its related quadratic equation, you replace y with 0. y=ax2 + bx + c 0 = ax2 + bx + c

  4. One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros.

  5. Additional Example 1A: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 1 Write the related function. 2x2 – 18 = y,or y = 2x2+ 0x– 18 Step 2 Graph the function. ● • The axis of symmetry is x = 0. • The vertex is (0, –18). • Two other points (2, –10) and • (3, 0) • Graph the points and reflectthem • across the axis of symmetry. x = 0 ● (3, 0) ● ● (2, –10) ● (0, –18)

  6. Check2x2 – 18 = 0 2x2 – 18 = 0 2(3)2 – 18 0 2(–3)2 – 18 0 2(9) – 18 0 2(9) – 18 0 18 – 18 0 18 – 18 0   0 0 0 0 Additional Example 1A Continued Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 3 Find the zeros. The zeros appear to be 3 and –3. The solutions of 2x2 – 18 = 0 are 3 and –3. Substitute 3 and –3 for x in the original equation.

  7. 2x2 – 12x + 18 = 0 y = 2x2 – 12x + 18 Additional Example 1B: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. –12x + 18 = –2x2 Step 1 Write the related function. Step 2 Graph the function. Use a graphing calculator. Step 3 Find the zeros. The only zero appears to be 3. This means 3 is the only root of 2x2 – 12x + 18.

  8. Step 1 Write the related function. y = 2x2 + 4x + 3  (–3, 9)  (1, 9)  (–2, 3)  (0, 3)  (–1, 1) Additional Example 1C: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x2 + 4x = –3 Step 2 Graph the function. • The axis of symmetry is x = –1. • The vertex is (–1, 1). • Two other points (0, 3) and • (1, 9). • Graph the points and reflectthem • across the axis of symmetry.

  9. Additional Example 1C Continued Solve the equation by graphing the related function. 2x2 + 4x = –3 Step 3 Find the zeros. The function appears to have no zeros. The equation has no real-number solutions.

  10. In Your Notes! Example 1a Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 1 Write the related function. y = x2 + 8x+ 16 x = –4 Step 2 Graph the function. • The axis of symmetry is x = –4. • The vertex is (–4, 0). • The y-intercept is 16. • Two other points are (–3, 1) and • (–2, 4). • Graph the points and reflectthem • across the axis of symmetry. ● ● (–2 , 4) ● ● (–3, 1) ● (–4, 0)

  11. Check y =x2 + 8x + 16 0 (–4)2 + 8(–4) + 16 0 16 – 32 + 16  0 0 In Your Notes! Example 1a Continued Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 3 Find the zeros. The only zero appears to be –4. Substitute –4 for x in the quadratic equation.

  12. Step 1 Write the related function. y =x2 + 6x + 10 In Your Notes! Example 1b Solve the equation by graphing the related function. 6x + 10 = –x2 x = –3 Step 2 Graph the function. • The axis of symmetry is x = –3 . • The vertex is (–3 , 1). • The y-intercept is 10. • Two other points (–1, 5) and • (–2, 2) • Graph the points and reflectthem • across the axis of symmetry. (–1, 5) ● ● (–2, 2) ● ● ● (–3, 1)

  13. Step 3 Find the zeros. The function appears to have no zeros In Your Notes! Example 1b Continued Solve the equation by graphing the related function. x2 + 6x + 10 = 0 The equation has no real-number solutions.

  14. y = –x2 + 4 Step 2 Graph the function. Use a graphing calculator. In Your Notes! Example 1c Solve the equation by graphing the related function. –x2 + 4 = 0 Step 1 Write the related function. Step 3 Find the zeros. The function appears to have zeros at (2, 0) and (–2, 0).

  15. Finding the roots of a quadratic polynomial is the same as solving the related quadratic equation.

  16.    Additional Example 2A: Finding Roots of Quadratic Polynomials Find the roots of x2 + 4x + 3 Step 1 Write the related equation. 0 = x2 + 4x + 3 y = x2 + 4x + 3 Step 2 Write the related function. y = x2 + 4x + 3 Step 3 Graph the related function. (–4, 3) • The axis of symmetry is x = –2. • The vertex is (–2, –1). • Two other points are (–3, 0) • and (–4, 3) • Graph the points and reflectthem • across the axis of symmetry. (–3, 0)  (–2, –1)

  17. 0 = x2 + 4x + 3 Check 0 = x2 + 4x + 3 0(–3)2 + 4(–3) + 3 0(–1)2 + 4(–1) + 3 09 – 12 + 3 01 – 4 + 3   0 0 0 0 Additional Example 2A Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –3 and –1. This means –3 and –1 are the roots of x2 + 4x + 3.

  18. (2, –15)     (1, –18)  (–0.5, –20.25). Additional Example 2B: Finding Roots of Quadratic Polynomials Find the roots of x2 + x – 20 Step 1 Write the related equation. y = x2 + 4x – 20 0 = x2 + x – 20 Step 2 Write the related function. y = x2 + 4x – 20 Step 3 Graph the related function. • The axis of symmetry is x = – . • The vertex is (–0.5, –20.25). • Two other points are (1, –18) • and (2, –15) • Graph the points and reflectthem • across the axis of symmetry.

  19. 0 = x2 + x – 20 Check 0 = x2 + x – 20 0(–5)2– 5 – 20 042 + 4 – 20 025 – 5 – 20 016 + 4 – 20   0 0 0 0 Additional Example 2B Continued Find the roots of x2 + x – 20 Step 4 Find the zeros. The zeros appear to be –5 and 4. This means –5 and 4 are the roots of x2 + x – 20.

  20. (4, 3)    (5, 0)   (6, –1). Additional Example 2C: Finding Roots of Quadratic Polynomials Find the roots of x2 – 12x + 35 Step 1 Write the related equation. y = x2 – 12x + 35 0 = x2 – 12x + 35 Step 2 Write the related function. y= x2 – 12x + 35 Step 3 Graph the related function. • The axis of symmetry is x = 6. • The vertex is (6, –1). • Two other points (4, 3) and • (5, 0) • Graph the points and reflectthem • across the axis of symmetry.

  21. 0 = x2 – 12x + 35 Check 0 = x2 – 12x + 35 052 – 12(5) + 35 072 – 12(7) + 35 025 – 60 + 35 049 – 84 + 35   0 0 0 0 Additional Example 2C Continued Find the roots of x2 – 12x + 35 Step 4 Find the zeros. The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x2 – 12x + 35.

  22.  (–2, 0)    (–1, –2) (–0.5, –2.25). In Your Notes! Example 2a Find the roots of each quadratic polynomial. x2 + x – 2 y = x2 + x – 2 Step 1 Write the related equation. 0 = x2 + x – 2 Step 2 Write the related function. y = x2 + x – 2 Step 3 Graph the related function. • The axis of symmetry is x = –0.5. • The vertex is (–0.5, –2.25). • Two other points (–1, –2) and • (–2, 0) • Graph the points and reflectthem • across the axis of symmetry.

  23. 0 = x2 + x – 2 Check 0 = x2 + x – 2 0(–2)2 + (–2) – 2 012 + (1) – 2 04 – 2 – 2 01 + 1 – 2   0 0 0 0 In Your Notes! Example 2a Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –2 and 1. This means –2 and 1 are the roots of x2 + x – 2.

  24. (0, 1)   ( , 4) ( , 0). • The axis of symmetry is x = . • The vertex is ( , 0). • Two other points (0, 1) and • ( , 4) • Graph the points and reflectthem • across the axis of symmetry. In Your Notes! Example 2b Find the roots of each quadratic polynomial. 9x2 – 6x + 1 y = 9x2 – 6x + 1 Step 1 Write the related equation. 0 = 9x2 – 6x + 1  Step 2 Write the related function. y = 9x2 – 6x + 1 Step 3 Graph the related function. 

  25. 0 = 9x2 – 6x + 1 Check 09()2 – 6() + 1 01 – 2 + 1  0 0 In Your Notes! Example 2b Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be one zero at . This means that is the root of 9x2 – 6x + 1.

  26.   (1, 6)  • The axis of symmetry is x = . • The vertex is ( , ). • Two other points (1, 6) and • ( , ) • Graph the points and reflectthem • across the axis of symmetry. In Your Notes! Example 2c Find the roots of each quadratic polynomial. 3x2 – 2x + 5 y = 3x2 – 2x + 5 Step 1 Write the related equation. 0 = 3x2 – 2x + 5 Step 2 Write the related function. y = 3x2 – 2x + 5  Step 3 Graph the related function.

  27. In Your Notes! Example 2c Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be no zeros. This means that there are no real roots of 3x2 – 2x + 5.

More Related