1 / 10

Function III

Y-axis. y = x 2 - 2x - 8. axis. Y-axis. y = mx + b. b. X-axis. X-axis. -b m. 1. -2. 4. - 8. (1,-9). Function III. Functions: Domain and Range By Mr Porter. Definitions. Function:.

murray
Download Presentation

Function III

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. Y-axis y = x2 - 2x - 8 axis Y-axis y = mx + b b X-axis X-axis -b m 1 -2 4 - 8 (1,-9) Function III Functions: Domain and Range By Mr Porter

  2. Definitions Function: A function is a set of ordered pair in which no two ordered pairs have the same x-coordinate. Domain The domain of a function is the set of all x-coordinates of the ordered pairs. [the values of x for which a vertical line will cut the curve.] Range The range of a function is the set of all y-coordinates of the ordered pairs. [the values of y for which a horizontal line will cut the curve] Note: Students need to be able to define the domain and range from the equation of a curve or function. It is encourage that student make sketches of each function, labeling each key feature.

  3. Y-axis y = 3x + 6 4 X-axis 6 Y-axis 6 X-axis Every vertical line will cut 2x+3y =12. Every vertical line will cut y = 3x + 6. -2 Every horizontal line will cut 2x+3y =12 Every horizontal line will cut y = 3x + 6 Linear Functions • Any equation that can be written in the • General form ax + by + c = 0 • Standard form y = mx + b Sketching Linear Functions. Find the x-intercep at y = 0 And the y-intercept at x = 0. Examples a) y = 3x + 6 b) 2x + 3y = 12 x-intercept at y = 0 2x = 12 x = 6 y-intercept at x = 0 3y = 12 y = 4 x-intercept at y = 0 0 = 3x + 6 x = -2 y-intercept at x = 0 y = 6

  4. Y-axis Y-axis (a,b) (4,5) X-axis X-axis Y-axis x = 4 x = a x = -2 X-axis (-2,-6) Special Lines Examples Vertical Lines: x = a - these are not functions, as the first element in any ordered pair is (a, y) a) x = 4 Equation of a vertical line is: i) x = a ii) x - a = 0 Domain: x = 4 Range: all y in R Sketch b) x + 2 = 0 Domain: x = a Range: all y in R Domain: x = -2 Range: all y in R

  5. Y-axis (-5,3) y = 3 X-axis Y-axis (a,b) y = a X-axis Y-axis X-axis y = -6 (2,-6) Special Lines Examples Horizontal Lines: y = a - these are functions, as the first element in any ordered pair is (x, a) a) y = 3 Equation of a horizontal line is: i) y = a ii) y - a = 0 Domain: all x in R Range: y = 3 Sketch b) y + 6 = 0 Domain: all x in R Range: y = b Domain: all x in R Range: y = -6

  6. The five steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. If a is negative, the parabola is concave down. 2) To find the y-intercept, put x = 0. 3) To find the x-intercept, form a quadratic and solve ax2 + bx + c = 0 * factorise * quadratic formula 4) Find the axis of symmetry by 5) Use the axis of symmetry x-value to find the y-value of the vertex, h 4) Axis of symmetry at = -1 Y-axis X-axis -3 -1 1 -3 (-1,-4) Parabola: y = ax2 +bx + c Example Sketch y = x2 + 2x - 3, hence, state its domain and range. 1) For y = ax2 + bx + c a = 1, b = +2, c = -3 . Concave-up a = 1 2) y-intercept at x = 0, y = -3 3) x-intercept at y = 0, (factorise ) (x - 1)(x + 3) = 0 x = +1 and x = - 3. 5) y-value of vertex: y = (-1)2 +2(-1) - 3 y = -4 Domain: all x in R Range: y ≥ -4 Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0

  7. The five steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. If a is negative, the parabola is concave down. 2) To find the y-intercept, put x = 0. 3) To find the x-intercept, form a quadratic and solve ax2 + bx + c = 0 * factorise * quadratic formula 4) Find the axis of symmetry by 5) Use the axis of symmetry x-value to find the y-value of the vertex, h 4) Axis of symmetry at = +2 Y-axis X-axis 2 (2,-1) -5 Parabola: y = ax2 +bx + c Example Sketch y = –x2 + 4x - 5, hence, state its domain and range. 1) For y = ax2 + bx + c, a = -1, b = +4, c = -5. Concave-down a = -1 2) y-intercept at x = 0, y = -5 3) x-intercept at y = 0, NO zeros by Quadratic formula. 5) y-value of vertex: y = -(2)2 +4(2) - 5 y = -1 Domain: all x in R Range: y ≤ -1 Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0

  8. Y-axis axis y = x2 - 2x - 8 The five steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. If a is negative, the parabola is concave down. 2) To find the y-intercept, put x = 0. 3) To find the x-intercept, form a quadratic and solve ax2 + bx + c = 0 * factorise * quadratic formula Find the axis of symmetry by 5) Use the axis of symmetry x-value to find the y-value of the vertex, h X-axis 1 -2 4 - 8 (1,-9) Worked Example 1: Your task is to plot the key features of the given parabola, sketch the parabola, then state clearly its domain and range. Sketch the parabola y = x2 - 2x - 8, hence state clearly its domain and range. Step 1: Determine concavity: Up or Down? For the parabola of the form y = ax2 + bx + c a = 1 => concave up Step 2: Determine y-intercept. Let x = 0, y = -8 Step 3: Determine x-intercept. Solve: x2 - 2x - 8 = 0 Factorise : (x - 4)(x + 2) = 0 ==> x = 4 or x = -2. Step 4: Determine axis of symmetry. Step 5: Determine maximum or minimum y-value (vertex). Substitute the value x = 1 into y = x2 - 2x - 8. Domain all x in R Range y ≥ -9 y = (1)2 - 2(1) - 8 = -9 Vertex at (1, -9)

  9. Y-axis The five steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. If a is negative, the parabola is concave down. 2) To find the y-intercept, put x = 0. 3) To find the x-intercept, form a quadratic and solve ax2 + bx + c = 0 * factorise * quadratic formula Find the axis of symmetry by 5) Use the axis of symmetry x-value to find the y-value of the vertex, h axis (1,16) 15 X-axis -5 -1 3 f(x)=15 - 2x - x2 Worked Example 2: Your task is to plot the key features of the given parabola, sketch  the parabola, then state clearly its domain and range. Sketch the parabola f(x) = 15 - 2x - x2, hence state clearly its domain and range. Step 1: Determine concavity: Up or Down? For the parabola of the form f(x) = ax2 + bx + c a = -1 => concave down Step 2: Determine y-intercept. Let x = 0, f(x) = +15 Step 3: Determine x-intercept. Solve: 15 - 2x - x2 = 0 Factorise : (3 - x)(x + 5) = 0 ==> x = 3 or x = -5. Step 4: Determine axis of symmetry. Step 5: Determine maximum or minimum y-value (vertex). Substitute the value x = -1 into y = 15 - 2x - x2. Domain: all x in R Range: y ≤ 16 y = 15 - 2(-1) - (-1)2 = 16 Vertex at (1, 16)

  10. Y-axis h(x) = 2x2 + 7x - 15 Y-axis 5 X-axis X-axis 11/2 -5 -13/4 21/2 -15 f(x) = 5 - 2x (-13/4 ,-211/8 ) h(x) = x2 + 2x + 5 Y-axis Y-axis 4 X-axis NO x-intercepts. (try quadratic formula?) 5 -4/5 (-1,4) -1 X-axis g(x) = 5x + 4 Exercise: For each of the following functions: a) sketch the curve b) sate the largest possible domain and range of the function. (i) f(x) = 5 - 2x (ii) h(x) = 2x2 + 7x - 15 Domain: All x in R Range: All y in R Domain: All x in R Range: All y ≥ -211/8 (iii) h(x) = x2 + 2x + 5 (iv) g(x) = 5x + 4 Domain: All x in R Range: All y ≥ 4 Domain: All x in R Range: All y in R

More Related