Math 1330 Section 2.1 Section 2.1: Linear and Quadratic Functions A linear function is a function that can be written in the form = + (m = slope and b = y-intercept). Its graph is a straight line. Recall: If a line passes through, and, its slope m is given by = ="rise" "run" The sign of the slope indicates whether the line is an increasing or a decreasing function. The slope of any horizontal line is 0 and the slope of any vertical line is undefined. Two Forms for an Equation of a Line: 1. = + (slope-intercept form) 2. =m x (point-slope form) Note: We are working with functions, so we want y on the left-hand side by itself. This means we need to rearrange our equations into slope-intercept form so we can write = + Example 1: Write an equation for a linear function whose x-intercept is 4 and the x-intercept for the inverse is 2. 1
Math 1330 Section 2.1 Example 2: Suppose the point (1, 2 is on the graph of a linear function and the point 4,3 is on the inverse function s graph. What is the equation for both the inverse and the function? Parallel Lines and Perpendicular Lines Two lines are parallel if and only if their slopes are the same. = Two lines are perpendicular if and only if their slopes are negative reciprocals of each other. = 1 or also = 1 Example 3: Write the equation of the linear function that passes through the point 4,5 and is parallel to the line 4 +3 =7. Example 4: Find the linear function such that 2 =5 and the graph of is perpendicular to the line = 2
Math 1330 Section 2.1 Quadratic Functions A quadratic function is a function which can be written in the form = + +, where a, b, and c are real numbers and a is not equal to zero. The Standard Form of a Quadratic Function Every quadratic function also known as a parabola is written as = + + or can be written in standard form: = h +. The vertex is the point h,. The axis of symmetry is the equation x = h. For a quadratic function, = + + or = h + The graph opens up if a > 0. If >1, is the parabola is narrower; if <1 the parabola is wider. The vertex is the turning point of the parabola. If the parabola opens upward the function has a minimum value (y-value). If the parabola opens downward the function has a maximum value (y-value). The axis of symmetry is a line through the vertex that divides the graph in half. The vertex of a parabola whose equation is = + + is, 2 2 And the axis of symmetry is = 3
Math 1330 Section 2.1 You should be able to identify the following: Direction the graphs opens(upwards or downwards) Whether the function has a maximum or a minimum y-intercept coordinates of the vertex equations of the axis of symmetry maximum or minimum Example 5: Write = 2 +12 15 in standard form. Find the a. vertex b. axis of symmetry c. maximum or minimum value Example 6: Sketch the graph of =3 +6 +7 by finding the six features of the function. 4
Math 1330 Section 2.1 Example 7: Find the quadratic function such that the axis of symmetry is = 2, the y- intercept is -6 and there is only one x-intercept. 5
Section 2.2 Polynomial Functions Definition: A polynomial function is a function which can be written in the form The numbers are called the coefficients of the polynomial function and. is the largest variable term of the polynomial function of x of degree n. The number is the coefficient of the variable to the highest power, is called the leading coefficient. Note: The variable is only raised to positive integer powers no negative or fractional exponents. However, the coefficients may be any real numbers, including fractions or irrational numbers like. The domain of any polynomial function is all real numbers. Example 1: Given leading coefficient?. What is the degree and Facts about polynomials: They are smooth curves, with no jumps or sharp points. A polynomial has at most turning points. A polynomial has at most n x-intercepts. A polynomial has exactly one y-intercept. End Behavior of Polynomial Functions The behavior of a graph of a function to the far left or far right is called its end behavior. Even Degree Positive Leading Coefficient Negative Leading Coefficient 1
Odd Degree Positive Leading Coefficient Negative Leading Coefficient Power functions A power function is a polynomial that takes the form, where n is a positive integer. Modifications of power functions can be graphed using transformations. Even-degree power functions: Odd-degree power functions: Note: Multiplying any function by a will multiply all the y-values by a. The general shape will stay the same. Example 2: Sketch the graph of. 2
Zeros of polynomials If f is a polynomial and c is a real number for which root of. then c is called a zero of, or a If c is a zero of f, then c is an x-intercept of the graph of. is a factor of. So if we have a polynomial in factored form, we know all of its x-intercepts. every factor gives us an x-intercept. every x-intercept gives us a factor. Note: In factoring the equation for the polynomial function, if the same factor x c occurs k times, we call c a repeated zero with multiplicity k. Description of the Behavior at Each x-intercept 1. Even Multiplicity: The graph touches the x-axis, but does not cross it (looks like a parabola there). 2. Odd Multiplicity of 1: The graph crosses the x-axis (looks like a line there). 3. Odd Multiplicity greater than or equal to 3: The graph crosses the x-axis and it looks like a cubic there. Steps to graphing other polynomials 1. Determine the leading term. Is the degree even or odd? Is the leading coefficient positive or negative? Use the answers to both questions to determine the end behavior. 2. Find the y-intercept. 3. Factor the polynomial. 4. Plot the x-intercepts and y-intercept on the 2-dimensional plane. 5. Draw the graph, keeping in mind the multiplicity of each zero and the behavior of the graph surrounding each zero. Note: Your graph should be smooth with no sharp corners.without calculus or plotting lots of points, we don t have enough information to know how high or how low the turning points are. 3
Example 3: Sketch the graph of. Example 4: Sketch the graph of 4
Example 5: Sketch the graph of Example 6: Write the equation of the quartic function with y intercept 5 which is tangent to the x axis at the points and (2,0). 5
Example 7: Given the graph of a polynomial determine what the equation of that polynomial. 6