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Function II axis Y-axis Y-axis y = x2 - 2x - 8 y = mx + b b X-axis X-axis -b m 1 -2 4 -8 (1,-9) Functions: Domain and Range By Mr Porter 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. Linear Functions Any equation that can be written in the • General form ax + by + c = 0 • Standard form y = mx + b Examples a) y = 3x + 6 Y-axis b) 2x + 3y = 12 Y-axis x-intercept at y = 0 0 = 3x + 6 x = -2 y-intercept at x = 0 y= 6 Sketching Linear Functions. Find the x-intercep at y = 0 And the y-intercept at x = 0. y = 3x + 6 x-intercept at y = 0 2x = 12 x= 6 y-intercept at x = 0 3y = 12 y= 4 6 X-axis -2 4 X-axis 6 Every vertical line will cut 2x+3y=12. Every vertical line will cut y = 3x + 6. Every horizontal line will cut 2x+3y=12 6 Every horizontal line will cut y = 3x + 6 Domain : All x R , real numbers Domain : All x R , real numbers Range : All y R , real numbers Range : All y R , real numbers Special Lines Examples Equation of a vertical line is: i) x = a Y-axis ii) x - a = 0 (4,5) X-axis Domain: x = 4 Range: all y in R (a,b) Sketch a) x = 4 x=4 Vertical Lines: x = a - these are not functions, as the first element in any ordered pair is (a, y) Y-axis b) x + 2 = 0 Y-axis x = -2 x=a X-axis X-axis Domain: x = a Range: all y in R (-2,-6) Domain: x = -2 Range: all y in R Special Lines Examples Horizontal Lines: y = a - these are functions, as the first element in any ordered pair is (x, a) Equation of a horizontal line is: i) y = a Y-axis ii) y - a = 0 Sketch a) y = 3 Y-axis (-5,3) y=3 X-axis Domain: all x in R Range: y = 3 (a,b) y=a X-axis b) y + 6 = 0 Y-axis X-axis Domain: all x in R Range: y = b y = -6 Domain: all x in R Range: y = -6 (2,-6) Parabola: y = ax2 +bx + c 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 b 4) Find the axis of symmetry by x 2a 5) Use the axis of symmetry x-value to find the y-value of the vertex, h Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0 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. b (2) 4) Axis of symmetry at x = -1 2a 2(1) 5) y-value of vertex: y = (-1)2 +2(-1) - 3 y = -4 Y-axis X-axis -3 Domain: all x in R Range: y ≥ -4 -1 1 -3 (-1,-4) Parabola: y = ax2 +bx + c 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 b 4) Find the axis of symmetry by x 2a 5) Use the axis of symmetry x-value to find the y-value of the vertex, h Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0 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. 4) Axis of symmetry at x b (4) = +2 2a 2(1) 5) y-value of vertex: y = -(2)2 +4(2) - 5 y = -1 Y-axis X-axis 2 (2,-1) Domain: all x in R Range: y ≤ -1 -5 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. Y-axis axis The five steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. y = x2 - 2x - 8 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 X-axis * quadratic formula 4)Find the axis 1of symmetry by -2 4 b x 2a 5) Use the axis of symmetry x-value to find the y-value of -8 the vertex, h (1,-9) 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. b (2) 1 x Step 4: Determine axis of symmetry. 2(1) 2a Step 5: Determine maximum or minimum y-value (vertex). Domain all x in R Range y ≥ -9 Substitute the value x = 1 into y = x2 - 2x - 8. 2 y = (1) - 2(1) - 8 = -9 Vertex at (1, -9) 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 axis The five Y-axis steps in sketching a parabola function: 1) If a is positive, the parabola is concave up. If a is negative, the parabola is(1,16) concave down. 2) To find the y-intercept, 15 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 3 b-1 x 2a 5) Use the axis of symmetry x-value to find the y-value of the vertex, h f(x)=15 - 2x - x2 Domain: all x in R Range: y ≤ 16 Step 2: Determine y-intercept. Let x = 0, f(x) = +15 Step 3: Determine x-intercept. X-axis Solve: 15 - 2x - x2 = 0 Factorise : (3 - x)(x + 5) = 0 ==> x = 3 or x = -5. b (2) 1 x Step 4: Determine axis of symmetry. 2(1) 2a Step 5: Determine maximum or minimum y-value (vertex). Substitute the value x = -1 into y = 15 - 2x - x2. 2 y = 15 - 2(-1) - (-1) = 16 Vertex at (1, 16) Exercise: For each of the following functions: a) sketch the curve b) sate the largest possible domain and range of the function. (ii) h(x) = 2x2 + 7x - 15 (i) f(x) = 5 - 2x Domain: All x in R Range: All y in R Y-axis Domain: All x in R Range: All y ≥ -211/8 Y-axis h(x) = 2x2 + 7x - 15 5 X-axis X-axis 11/2 -13/4 -5 21/2 -15 f(x) = 5 - 2x (iii) h(x) = x2 + 2x + 5 h(x) = x2 + 2x + 5 Y-axis (-13/4 ,-211/8 ) (iv) g(x) = 5x + 4 Domain: All x in R Range: All y ≥ 4 Domain: All x in R Range: All y in R Y-axis 4 X-axis 5 (-1 ,4) -1 NO x-intercepts. (try quadratic formula?) X-axis -4/5 g(x) = 5x + 4