Quadratic functions
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A polynomial function can be expressed as a sum of terms with non-negative integer exponents. A quadratic function is a polynomial of degree 2 in the form p(x)=ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola, which is symmetric about the axis of symmetry and has a unique vertex. Changing the coefficients a, b, and c changes the position, shape, and orientation of the parabola.
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2) Properties and equations of parabolas, circles, ellipses, and hyperbolas including their standard forms and key components.
3) Instructions for using a graphing calculator to graph functions and view graphs.
4) Information about the format of the upcoming test which will include 8-10 questions involving these graphing concepts.
Graphing quadratic equations
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
April 10, 2015
The document discusses graphing quadratic functions. It begins with reviewing key concepts like the vertex and axis of symmetry. The effects of the a, b, and c coefficients on the parabola are explained. Examples are provided to show how changing these values affects the width, direction opened, and vertical translation of the graph. The class will graph various quadratic functions by finding the axis of symmetry, vertex, y-intercept, and other points to plot the parabola. Students are assigned class work problems to graph quadratic functions and show their work.
Parabola 091102134314-phpapp01
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
WEEK-4-Piecewise-Function-and-Rational-Function.pptx
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
Quadraticfunctionpresentation 100127142417-phpapp02
The document defines quadratic functions and discusses their various forms, including general, vertex, and factored forms. It also covers solving quadratic equations using methods like the quadratic formula, factoring, and completing the square. Additionally, it discusses key features of quadratic graphs like x-intercepts, y-intercepts, the vertex, and concavity. Examples are provided to illustrate finding these features and graphing parabolas.
Lecture 2 family of fcts
This document provides an overview of various mathematical models and functions, including:
1) Linear, polynomial, power, rational, algebraic, trigonometric, exponential, and logarithmic functions.
2) It describes the key properties of each type of function, such as their domains, ranges, and graph shapes.
3) It also discusses how functions can be transformed through stretching, shrinking, shifting, and reflection.
Quadratic equation.pptx
- The document discusses revising and sketching parabolic functions of the form y=ax+b, including how the parameters a and b affect the graph shape and position.
- It introduces parabolic equations in standard form y=a(x-p)+q and turning point form, identifying characteristics like the turning point, axes of symmetry, intercepts, and asymptotes.
- Examples are provided to demonstrate how to determine these characteristics, sketch the graph, and state the domain and range for parabolic functions given in equation form.
Graph a function
This document provides an overview of various types of functions and their graphs. It begins with linear functions of the form y=mx+c and discusses how shifting these functions along the x- or y-axis changes their graphs. It then covers quadratic, square root, cube, reciprocal, constant, identity and absolute value functions. Piecewise, polynomial, algebraic, and transcendental functions are also defined. The document discusses bounded vs unbounded functions and concludes by examining circular and hyperbolic functions and their graphs.
Fst ch2 notes
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Quadratic functions
- 2. PolyNomials Definition:A polynomial function is a function that can be expressed in the form: p(x) = an xn + an-1 xn-1 + … + a1 x + a0 Where an , an-1 , … , a1 , a0 are real numbers, an 0,≠ the exponents are non-negative integers Definition:The degree of a polynomial is largest exponent of x. ✔ ✔ ✔ ✔ The degree is 2 The degree is 1 The degree is 0 The degree is 3
- 3. Quadratic functions A polynomial of degree 0 is called a constant function. A polynomial of degree 1 is called a linear function. Definition:A degree 2 polynomial function is called a quadratic function. The general form a quadratic function is p(x) = ax2 + bx + c where a, b, and c are real numbers with a 0≠ . Quadratic functions are incredibly important functions that show up everywhere in the real world.
- 4. The graph of a quadratic polynomial is called a parabola. Parabolas p(x) = ax2 + bx + c a > 0 a < 0 vertex vertex Axis of Symmetry Axis of Symmetry
- 5. How does the graph of a quadratic function change as we change a, b, and c? Parabolas a decreases from 1 towards 0
- 6. How does the graph of a quadratic function change as we change a, b, and c? Parabolas a increases from 1 to 10
- 7. How does the graph of a quadratic function change as we change a, b, and c? Parabolas c increases from 0 to 2
- 8. How does the graph of a quadratic function change as we change a, b, and c? Parabolas c decreases from 0 to -2
- 9. Definition:The standard form of a quadratic function is p(x) = a(x – h)2 + k Where (h, k) is the vertex of its graph and a 0≠ . Standard form
- 10. Summary General Form: Vertex: Axis of symmetry: standard Form: Vertex: Axis of symmetry: Parabola opens up Parabola opens down
- 11. problems Find the vertex and the x-intercepts of the following functions:
- 12. problems Find the quadratic function with the indicated vertex that passing though the given point: 1. Vertex: (2,3) Point: (0,2) 2. Vertex: (-2,-2) Point: (-1,0) 3. Vertex: (6,6) Point: (1/2, 3/4)
- 13. problems The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) that the company spends on advertising according to the model: P(x) = 230 + 20x – 0.5x2 How much should the company spend on advertising to maximize profits?