Download presentation

Presentation is loading. Please wait.

1
**Functions: Domain and Range By Mr Porter**

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

2
**Definitions Function: Domain Range**

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
**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 Y-axis X-axis 4 6 -2 6 Y-axis X-axis y = 3x + 6 Every vertical line will cut 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 Every vertical line will cut y = 3x + 6. Every horizontal line will cut 2x+3y =12 Every horizontal line will cut y = 3x + 6

4
**Special Lines Examples Vertical Lines: x = a**

(4,5) x = 4 X-axis Y-axis 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 (a,b) x = a X-axis Y-axis Range: all y in R Sketch b) x + 2 = 0 (-2,-6) x = -2 X-axis Y-axis Domain: x = a Range: all y in R Domain: x = -2 Range: all y in R

5
**Special Lines Examples Horizontal Lines: y = a**

(-5,3) y = 3 X-axis Y-axis 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 (a,b) y = a X-axis Y-axis Range: y = 3 Sketch b) y + 6 = 0 (2,-6) y = -6 X-axis Y-axis Domain: all x in R Range: y = b Domain: all x in R Range: y = -6

6
**Parabola: y = ax2 +bx + c Example**

Sketch y = x2 + 2x - 3, hence, state its domain and range. 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 1) For y = ax2 + bx + c a = 1, b = +2, c = 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. 4) Axis of symmetry at = -1 5) y-value of vertex: y = (-1)2 +2(-1) - 3 y = -4 -3 1 (-1,-4) X-axis Y-axis -1 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
**Parabola: y = ax2 +bx + c Example**

Sketch y = –x2 + 4x - 5, hence, state its domain and range. 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 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. 4) Axis of symmetry at = +2 5) y-value of vertex: y = -(2)2 +4(2) - 5 y = -1 (2,-1) -5 X-axis Y-axis 2 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
**Step 1: Determine concavity: Up or Down?**

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. -2 4 (1,-9) axis 1 X-axis Y-axis Step 1: Determine concavity: Up or Down? 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 For the parabola of the form y = ax2 + bx + c a = 1 => concave up y = x2 - 2x - 8 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. - 8 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
**Step 1: Determine concavity: Up or Down?**

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) = x - x2, hence state clearly its domain and range. -5 3 (1,16) axis -1 X-axis Y-axis Step 1: Determine concavity: Up or Down? 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 For the parabola of the form f(x) = ax2 + bx + c a = -1 => concave down Step 2: Determine y-intercept. f(x)=15 - 2x - x2 15 Let x = 0, f(x) = +15 Step 3: Determine x-intercept. Solve: x - 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 = x - x2. Domain: all x in R Range: y ≤ 16 y = (-1) - (-1)2 = 16 Vertex at (1, 16)

10
**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 -5 11/2 -13/4 Y-axis X-axis h(x) = 2x2 + 7x - 15 -15 (-13/4 ,-211/8 ) X-axis Y-axis f(x) = 5 - 2x 5 21/2 (iii) h(x) = x2 + 2x + 5 (iv) g(x) = 5x + 4 NO x-intercepts. (try quadratic formula?) -1 Y-axis X-axis h(x) = x2 + 2x + 5 5 (-1 ,4) Domain: All x in R Range: All y ≥ 4 Domain: All x in R Range: All y in R X-axis Y-axis g(x) = 5x + 4 4 -4/5

Similar presentations

Presentation is loading. Please wait....

OK

Graphing Quadratic Equations

Graphing Quadratic Equations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on e-sales order processing system Ppt on network theory tutorials Ppt on corporate grooming etiquette Ppt on review writing software Ppt on swami vivekananda Ppt on surface area and volume of frustum Show ppt on second monitor Ppt on layer 3 switching vs layer Ppt on internal auditing process approach Ppt on 360 degree performance appraisal