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Function III Functions: Domain and Range By Mr Porter -24 (1,-9) axis 1 X-axis Y-axis y = x 2 - 2x - 8 - 8 X-axis Y-axis y = mx + b b -b m

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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.

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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 + 6b) 2x + 3y = 12 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 y = 3x + 6 x-intercept at y = 0 2x = 12 x =6 y-intercept at x = 0 3y =12 y =4 -2 6 Y-axis X-axis y = 3x + 6 Y-axis X-axis 4 6 Every vertical line will cut 2x+3y =12. Every horizontal line will cut 2x+3y =12

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

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

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Parabola: y = ax 2 +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 ax 2 + 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 Example Sketch y = x 2 + 2x - 3, hence, state its domain and range. 1) For y = ax 2 + 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. 4) Axis of symmetry at = -1 5) y-value of vertex: y = (-1) 2 +2(-1) - 3 y = -4 Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0 Domain: all x in R Range: y ≥ -4 -3 1 (-1,-4) -3 X-axis Y-axis

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Parabola: y = ax 2 +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 ax 2 + 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 Example Sketch y = –x 2 + 4x - 5, hence, state its domain and range. 1) For y = ax 2 + 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 Domain: all x in R Range: y ≤ -1 (2,-1) -5 X-axis Y-axis 2 Domain: all x in R Range: y ≥ h for a > 0 Range: y ≤ h for a < 0

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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 = x 2 - 2x - 8, hence state clearly 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 ax 2 + 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 Step 1: Determine concavity: Up or Down? For the parabola of the form y = ax 2 + bx + c a = 1 => concave up Step 2: Determine y-intercept. Let x = 0, y = -8 Step 3: Determine x-intercept. Solve: x 2 - 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 = x 2 - 2x - 8. y = (1) 2 - 2(1) - 8 = -9Vertex at (1, -9) Domain all x in R Range y ≥ -9 -24 (1,-9) axis 1 X-axis Y-axis y = x 2 - 2x - 8 - 8

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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 - x 2, hence state clearly its domain and range. Step 1: Determine concavity: Up or Down? For the parabola of the form f(x) = ax 2 + 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 - x 2 = 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 - x 2. y = 15 - 2(-1) - (-1) 2 = 16Vertex at (1, 16) Domain: all x in R Range: y ≤ 16 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 ax 2 + 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 -5 3 (1,16) axis X-axis Y-axis f(x)=15 - 2x - x 2 15

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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) = 2x 2 + 7x - 15 (iv) g(x) = 5x + 4(iii) h(x) = x 2 + 2x + 5 X-axis Y-axis f(x) = 5 - 2x 5 21/221/2 Domain: All x in R Range: All y in R -5 11/211/2 -1 3 / 4 Y-axis X-axis h(x) = 2x 2 + 7x - 15 -15 (-1 3 / 4,-21 1 / 8 ) Domain: All x in R Range: All y ≥ -21 1 / 8 NO x-intercepts. (try quadratic formula?) Y-axis X-axis h(x) = x 2 + 2x + 5 5 (-1,4) Domain: All x in R Range: All y ≥ 4 X-axis Y-axis g(x) = 5x + 4 4 -4/5-4/5 Domain: All x in R Range: All y in R

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