Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

10.3

530 views

Published on

Published in: Education, Technology
  • Be the first to comment

  • Be the first to like this

10.3

  1. 1. y = a(x - p)(x-q)
  2. 2. Characteristics of  The general form where is called the intercept form of a quadratic function.  The x-intercepts are p and q.  The axis of symmetry is halfway between and AoS:  The parabola opens up if and down if y = a(x- p)(x-q) a ¹ 0 y = a(x - p)(x-q) (p,0) (q,0) x = p+q 2 a > 0 a < 0
  3. 3. Example 1 Find x-intercepts Find the x-intercepts of the graph of ( )1+x= ( ).5x–y – SOLUTION To find the x-intercepts, you need to find the values of x when y = 0. Write original function.=y ( )1+x ( )5x– – Substitute 0 for y.=0 ( )1+x ( )5x– – Zero-product property1+x 5x – = 0= 0 or Solve for x.x 5x == or1– The x-intercepts are and 5.1–ANSWER
  4. 4. Example 2 Graph a quadratic function in intercept form Graph ( )1 +x= ( ).3x–y –2 STEP 2 Find and draw the axis of symmetry: 2 q+p x = = 2 +1 ( )3– = 1.– STEP 1 Identify the x-intercepts. The x-intercepts are and Plot (1, 0) and ( 3, 0). p = 1 q = –3. –
  5. 5. Example 2 Graph a quadratic function in intercept form STEP 4 Draw a parabola through the vertex and the points where the x-intercepts occur. So, the vertex is ( , 8).–1 STEP 3 Find and plot the vertex. The axis of symmetry is x , so the x-coordinate of the vertex is . To find the y-coordinate of the vertex, substitute for x and simplify. = –1 –1 –1 ( )1 += ( )3–y –2 1– 1– 8=
  6. 6. Example 3 Graph a quadratic function in standard form Graph y = 12 +x2 12.x–3 Rewrite the quadratic function in intercept form. STEP 1 Write the function.=y 12 +x2 12x–3 Factor out 3.=y 4 +x2 4x–3( ) = x 2–3( )2 Factor the trinomial. Write in intercept form.= x 2–3( ) x 2–( ) STEP 2 Identify the x-intercepts. There is one x-intercept, 2. Plot (2, 0).
  7. 7. Example 3 STEP 3 Find and draw the axis of symmetry: 2 q+p x = = = 2. 2 2+2 STEP 4 Find and plot the vertex. The axis of symmetry is x 2, so the x-coordinate of the vertex is 2, which is also the x-intercept. So, the vertex is (2, 0). = Graph a quadratic function in standard form
  8. 8. Example 3 Draw a parabola through the points. STEP 6 STEP 5 Plot a point and its reflection. Choose a value for x, say x 1. When x 1, y 3. Plot (1, 3). By reflecting the point in the axis of symmetry, you can also plot (3, 3). = = = Graph a quadratic function in standard form
  9. 9. Example 4 Write a quadratic function in intercept form Write a quadratic function in intercept form whose graph has x-intercepts 1 and 3 and passes through the point (0, 12). – – STEP 1 Substitute the x-intercepts into The x-intercepts are p ( )px= ( ).y –a qx – and= 1– q 3.= ( )1x= ( )y +a 3x – Simplify. –Substitute 1 for p and 3 for q.( x= ( )y –a 3x –( )1– )
  10. 10. Example 4 Write a quadratic function in intercept form Find the value of a in using the given point (0, 12). STEP 2 ( )1x= (y +a 3x – ) – Simplify.12– = 3a– Divide each side by 3.–4 = a ANSWER y =The function in intercept form is ( )1x (+4 3x – ). Substitute 0 for x and 12 for y.–( )10= ( )+ 30 –12– a
  11. 11. Example 5 Model a parabolic path using intercept form BIOLOGY When a dolphin leaps out of the water, its body follows a parabolic path through the air. Write a function whose graph is the path of the dolphin in the air. SOLUTION STEP 1 ( )0x= (y a 4x – ),– = (y ax 4x – ).or Identify the x-intercepts. The dolphin leaves the water at (0, 0) and re- enters at (4, 0), so the x-intercepts of the path are p 0 and q 4. The function is of the form = =
  12. 12. Example 5 Model a parabolic path using intercept form STEP 2 Find the axis of symmetry and the vertex. The axis of symmetry is The maximum height of 2 meters occurs on the axis of symmetry, so the vertex of the graph is (2, 2).2 4+0 x = 2.= Find the value of a. Substitute the coordinates of the vertex into the function. The vertex is (2, 2), so 2 a (2 – 0)(2 – 4), or a STEP 3 = = 2 1 – .
  13. 13. Example 5 Model a parabolic path using intercept form ANSWER The graph of the function path. is the dolphin’s 2 1 = – (x 4x – )y
  14. 14. 10.3 Warm-Up Find the x-intercepts of the graph of the quadratic function. 1. Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercept(s). 2. Write a quadratic function in intercept form whose graph has the given x-intercept(s) and passes through the given point. 3. x-intercepts: -6 and 2. point: y = -2(x -5)(x +1) y = (x+3)(x-4) (-2,8)

×