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# Note for MATH 1330 with Professor Flagg at UH

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15

PreCalSection21 Page 1 of7 Math 1330 Section 21 Linear and Quadratic Functions A linear function is one that has the form y mxb Where in and b are constants The graph of a linear function is a line With slope in and yintercept b The sign of the slope indicates Whether the line is an increasing or a decreasing function mlt0 mgt0 m 0 is a horizontal line in unde ned is a vertical line PreCalSecti0n2l Page 2 of7 Example 1 Graph the equation using its slope and intercept y 3x 2 Example 2 Graph f x 2x 3 Parallel Lines Two lines are parallel if they never cross In terms of equations two lines are parallel if they have the same slope Perpendicular Lines Two lines are perpendicular if they meet at a right angle In terms of equations two lines with slopes m1 and m2 are perpendicular if and only if mlm2 l Formulas f0r Lines Slope Given points qy1 and x2 y2 the slope of the line passing yz yr x2 x1 I through these points is m PointSlope Given a point qy1 and a slope m the equation for the line that passed through the point having slope m is y y1 mx x1 SlopeIntercept The slopeintercept equation for a line is an equation of the form y mx b where m is the slope and b is the yintercept PreCalSecti0n2l Page 3 of7 Writing Eguations of Lines Example 3 Write the equation of the linear function f such that f 2 1 and f3 2 Example 4 Write the equation of the linear function fx that passes through the point 45 and is parallel to the line 4x3 y 7 Example 5 Find the linear function f that is perpendicular to the line containing 42 and 104 and passes through the midpoint of the line segment connecting these points Example 6 Find the linear function f that passes through the point 36 and such that f 1 8 9 PreCalSecti0n2l Page 4 of7 Quadratic Functions Definition A quadratic function f is a function that can be put in the form fx ax2 bxc where a at 0 and a b and c are constants Graphing a Quadratic Function Standard Form fx ax h2 k The graph of f is a parabola The axis of symmetry is the vertical line xk The direction the parabola open depends on the sign of the constant a If a gt 0 then the parabola opens up If a lt 0 then the parabola opens down Graphing a quadratic equation procedure 1 Put the function into standard form by completing the square 2 Identify the vertex 3 Use the transformation rules to transform the graph of y x2 into the given function Maximum or Minimum Value of a Quadratic Function A quadratic function has either a maximum or minimum value depending on whether the parabola opens up or down This max or min is the Y VALUE of the vertex If the coefficient of the xsquared term is positive the parabola opens up and the vertex is a MINIMUM If the coefficient of the xsquared term is negative the parabola opens down and the vertex is a MAXIMUM PreCalSecti0n2l Page 5 of7 Example 7 Write the quadratic equation in standard form by completing the square Identify the vertex and the axis of symmetry and the maximum or minimum value Then sketch the graph fx 2x2 4xl fx 3x2 6x l PreCalSection2l Page 6 of7 A Formula for Finding the Vertex A quadratic function in the form f x ax h2 k Formula the Vertex is the Point f 2a 2a Example 8 Let f x x2 3x3 Find the vertex and maximum or minimum Finding the Intercepts of the Graph of a Quadratic Function YIntercept is just the value of the function at x 0 The xintercepts are the values of x such that fx0 To find them we need to solve the equation fx0 Methods for solving f x axz bx c l factoring 2 completing the square 3 the quadratic formula The uadratic Formula The solutions to the equation f x ax2 bx c are the values b i I b2 4610 2a The expression 12 4ac is called the discriminant If b2 4616 gt 0 then the quadratic has 2 real roots two xintercepts x If b2 4610 lt 0 then the quadratic equation has no real roots it does NOT cross the x axis at all If b2 4616 0 then the quadratic has just one real root ie one xintercept The intercept is the vertex of the parabola which geometrically means the parabola sits on the x axis and either opens up or down PreCalSecti0n2l Page 7 of7 Example 9 For the following quadratic functions find a the vertex b the axis of symmetry the line x x value of the vertex 0 the maximum or minimum value 1 the yintercept e the xintercept t sketch the graph fx3x2 6x5 fx x2 8xl6

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