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A Discussion of Diﬀerent Functions Mathematics 4 June 27, 2012Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 1 / 14
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Linear Functionsf (x) = mx + bLinear Function A linear function has the form f (x) = mx + b where m is the slope and b is the y-intercept. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 2 / 14
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Linear Functionsf (x) = mx + bLinear Function A linear function has the form f (x) = mx + b where m is the slope and b is the y-intercept. The domain of a linear function is {x | x ∈ R} Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 2 / 14
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Linear Functionsf (x) = mx + bLinear Function A linear function has the form f (x) = mx + b where m is the slope and b is the y-intercept. The domain of a linear function is {x | x ∈ R} The range is {y | y ∈ R} Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 2 / 14
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Linear Functionsf (x) = mx + bLinear Function f (x) = f −1 (x) = Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 3 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function A quadratic function has the form f (x) = ax2 + bx + c where a, b, c ∈ R, a = 0. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 4 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function A quadratic function has the form f (x) = ax2 + bx + c where a, b, c ∈ R, a = 0. The graph of a quadratic function is a parabola. The graph opens up if a > 0 and opens down when a < 0. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 4 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function A quadratic function has the form f (x) = ax2 + bx + c where a, b, c ∈ R, a = 0. The graph of a quadratic function is a parabola. The graph opens up if a > 0 and opens down when a < 0. The vertex of a parabola is given by the vertex equation −b −b ,f . 2a 2a Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 4 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function A quadratic function has the form f (x) = ax2 + bx + c where a, b, c ∈ R, a = 0. The graph of a quadratic function is a parabola. The graph opens up if a > 0 and opens down when a < 0. The vertex of a parabola is given by the vertex equation −b −b ,f . 2a 2a The vertex can also be determined by using completing the square and transforming the equation into the vertex form of the quadratic equation: (y − k) = a (x − h)2 . Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 4 / 14
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Quadratic FunctionsExample: Find the vertex (use completing the square), zeros, and graph of f (x) = −2x2 + 8x − 5: Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 5 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function The zeros of a quadratic function can be solved by letting f (x) = 0 and solving for x. These are also the x-intercepts of the graph. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 6 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function The zeros of a quadratic function can be solved by letting f (x) = 0 and solving for x. These are also the x-intercepts of the graph. The domain of a quadratic function is {x | x ∈ R}. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 6 / 14
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Quadratic Functionsf (x) = ax2 + bx + cQuadratic Function The zeros of a quadratic function can be solved by letting f (x) = 0 and solving for x. These are also the x-intercepts of the graph. The domain of a quadratic function is {x | x ∈ R}. The range is {y | y ≥ k} if the graph opens up, and {y | y ≤ k} when the graph opens down. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 6 / 14
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Quadratic FunctionsExample:Find the vertex, zeros, domain, range and graph of f (x) = 3x2 + 3x + 2. Identify the interval for which the graph is increasing and decreasing: Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 7 / 14
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Quadratic FunctionsExample: Given the function f (x) = 2x2 whose graph is shown below: 1 Modify the function such that the graph will move 2 units up. 2 Modify the new function such that the graph will move 3 units to the left. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 8 / 14
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Absolute Value Functionsf (x) = a |x − h| + kAbsolute Value Function An absolute value function has the form f (x) = a |x − h| + k where a ∈ R, a = 0. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 9 / 14
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Absolute Value Functionsf (x) = a |x − h| + kAbsolute Value Function An absolute value function has the form f (x) = a |x − h| + k where a ∈ R, a = 0. The graph of an absolute value function forms the shape of a V. The graph opens up if a > 0 and opens down when a < 0. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 9 / 14
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Absolute Value Functionsf (x) = a |x − h| + kAbsolute Value Function An absolute value function has the form f (x) = a |x − h| + k where a ∈ R, a = 0. The graph of an absolute value function forms the shape of a V. The graph opens up if a > 0 and opens down when a < 0. The slope of the legs of an absolute value function is given by both a and −a. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 9 / 14
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Absolute Value Functionsf (x) = a |x − h| + kAbsolute Value Function An absolute value function has the form f (x) = a |x − h| + k where a ∈ R, a = 0. The graph of an absolute value function forms the shape of a V. The graph opens up if a > 0 and opens down when a < 0. The slope of the legs of an absolute value function is given by both a and −a. The vertex of the graph of an absolute value function is given by the (h, k). Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 9 / 14
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Absolute Value FunctionsExample: Find the vertex, zeros, domain, range and graph of f (x) = 2 |x + 3| − 5. Identify the interval for which the graph is increasing and decreasing: Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 10 / 14
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Absolute Value FunctionsExample: Given the graph below of the previous function f (x) = 2 |x + 3| − 5, ﬁnd the equation of the function for the following cases: 1 The graph is moved two units to the left. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 11 / 14
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Absolute Value FunctionsExample: Given the graph below of the previous function f (x) = 2 |x + 3| − 5, ﬁnd the equation of the function for the following cases: 1 The graph is moved two units to the left. 2 The graph is then moved 4 units up. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 11 / 14
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Absolute Value FunctionsExample: Given the graph below of the previous function f (x) = 2 |x + 3| − 5, ﬁnd the equation of the function for the following cases: 1 The graph is moved two units to the left. 2 The graph is then moved 4 units up. 3 The direction of the graph is then inverted. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 11 / 14
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Absolute Value FunctionsExample: Given the graph below of the previous function f (x) = 2 |x + 3| − 5, ﬁnd the equation of the function for the following cases: 1 The graph is moved two units to the left. 2 The graph is then moved 4 units up. 3 The direction of the graph is then inverted. 4 The slopes of the legs are then reduced to 0.5 and −0.5. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 11 / 14
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The Square Root FunctionConsider the function f (x) = x2 , whose domain is {x | x ≥ 0}. f (x) = x2 , x ≥ 0 f −1 (x) =Find the inverse of this function both algebraically and graphically. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 12 / 14
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The Square Root Function √Given the square root function f (x) = x, whose graph is shown below: 1 Determine the domain and range. √ f (x) = x Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 13 / 14
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The Square Root Function √Given the square root function f (x) = x, whose graph is shown below: 1 Determine the domain and range. 2 Move the graph 2 units up. √ f (x) = x Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 13 / 14
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The Square Root Function √Given the square root function f (x) = x, whose graph is shown below: 1 Determine the domain and range. 2 Move the graph 2 units up. 3 Move the graph 3 units right. √ f (x) = x Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 13 / 14
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The Square Root Function √Given the square root function f (x) = x, whose graph is shown below: 1 Determine the domain and range. 2 Move the graph 2 units up. 3 Move the graph 3 units right. 4 Flip the graph horizontally. √ f (x) = x Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 13 / 14
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The Square Root Function √Given the square root function f (x) = x, whose graph is shown below: 1 Determine the domain and range. 2 Move the graph 2 units up. 3 Move the graph 3 units right. 4 Flip the graph horizontally. √ f (x) = x 5 Flip the graph vertically. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 13 / 14
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The Square Root FunctionGiven the graph of the square root function below, ﬁnd the equation ofthe function. Mathematics 4 () A Discussion of Diﬀerent Functions June 27, 2012 14 / 14
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