Quadratic Functions
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This document provides information about graphing parabolas on a graphing calculator. It discusses the key parts of a parabola graph like the vertex, axis of symmetry, zeros, and y-intercept. It explains how to input equations into the graphing calculator and use functions to find the vertex, zeros, and maximum or minimum points. The document also covers how changing variables in the standard parabola equation affects the shape, direction, and location of the graphed parabola. Some examples of word problems involving parabolas are presented as well.
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Determines the relationship between a rectangular prism and a pyramid
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1) Quadratic functions can be represented by the equation f(x) = ax^2 + bx + c and form a parabolic shape that can open up or down.
2) The axis of symmetry of a parabola passes through the vertex (highest/lowest point) and can be found using the equation x = -b/2a.
3) The vertex coordinates can be determined by substituting the axis of symmetry value into the original equation.
4) Quadratic functions can have 0, 1, or 2 x-intercepts depending on the number of real solutions to the related quadratic equation.
Developing Expert Voices
The student reflects on completing a math project for their calculus course as a way to study for an upcoming exam. They acknowledge that they procrastinated significantly but were able to cover a broad range of calculus concepts through multi-step word problems selected from different units. While the assignment did not dramatically increase their knowledge, it helped reinforce some details and connections between topics. The student resolves to select deadlines more wisely and stop procrastinating for future projects.
Solving Quadratics
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the definitions and formulas for each form. It then explains how to graph each form by identifying key features of the equation, finding important points like the vertex, axis of symmetry, intercepts, and connecting points to sketch the parabolic curve. Graphing techniques include using the value of a to determine the opening direction, using b and c for standard form, using h and k for vertex form, and using p and q for intercepts form.
Algebra 2. 9.16 Quadratics 2
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
Quadratics
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form - and how to graph each. It provides the key characteristics and steps to graph a quadratic function for each form. Standard form is ax2 + bx + c, vertex form is a(x-h)2 + k, and intercept form is a(x-p)(x-q). The document explains how to identify the vertex, axis of symmetry, intercepts, and whether the graph opens up or down based on the coefficients for each form. It also gives the process to follow to plot points and sketch the parabolic graph.
WRITING AND GRAPHING LINEAR EQUATIONS 1.pptx
The document discusses linear equations and graphing linear relationships. It defines key terms like slope, y-intercept, x-intercept, and provides examples of writing linear equations in slope-intercept form and point-slope form given certain information like a point and slope. It also discusses using tables and graphs to plot linear equations and find slopes from graphs or between two points. Real-world examples of linear relationships are provided as well.
Calc 7.1a
This document discusses calculating the area between two curves using integration. It explains that the area between curves f(x) and g(x) from x=a to x=b can be found by integrating f(x)-g(x) from a to b. It provides examples of finding the area between curves, including when the curves intersect multiple times, requiring splitting the region into separate intervals. The key steps are sketching the region, identifying intersection points, determining which curve is above the other in each interval, and setting up the correct integral expressions with the proper limits of integration. Representative rectangles are used to visualize the accumulation of area, whether vertical or horizontal depending on the problem.
Graphs of trigonometric exponential functions lecture
The document describes properties of trigonometric and exponential functions. It discusses:
1) The graphs of sine and cosine functions have a domain of real numbers, range from -1 to 1, and cycle through these values every 2π units of x with a smooth curve.
2) Exponential functions with a base greater than 1 have a domain of all real numbers, range from 0 to infinity, and increase rapidly with horizontal asymptote at y=0.
3) The amplitude and period of trigonometric functions can be adjusted through multiplication and composition with other functions.
Transformations
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
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Pretest
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Quadratic Functions
- 2. a(x - h)² + ky = ax² + bx + c a – width parabola b – vertical shift c – horizontal shift h – vertical shift k - horizontal
- 3. Graphs • Important Parts of the Graph – Vertex – Axis of Symmetry – Zeros (Intercepts) – Opening
- 4. Important Parts of the Graph • Vertex – The Min/Max point of the Parabola • Axis of Symmetry – Imaginary line that goes through the vertex. It splits the parabola into two mirror images • Zeros – Point(s) on the graph where it crosses the X- axis • Y-intercept – Point on the graph were it crosses the Y- axis • Opening
- 5. How to Use a Graphing Calculator • Buttons to know: – Y= – Graph – Trace – Window – Variable
- 6. What do the Buttons do? • Y= – Area where you input equation(s) of the graph • Graph – Shows the graph(s) • Window – Adjusts the graph’s min/max X and Y values • Trace – Able to run along the graph – Second Function (calculate): Is able to find the min/max value, zeros, etc. of graph • Variable – Easy access to variables
- 7. How to Input a Graph on the Calculator • Click “Y=” button to start • Input your equation(s) • Click the graph button to view your graph
- 8. How to find the Vertex of your Graph • 2nd Function > Trace • Select Min or Max depending on the opening of the graph • Select a point on the left side of the vertex then the right side of it, then take a guess at what it could be • Vertex will show up at the bottom of screen
- 9. How to find the Zero’s of the Graph • 2nd Function > Trace • Select Zero’s • Choose a point on the left side of a zero then a right (points must have different signs), lastly take a guess • Zero will show up at the bottom of screen • Usually needed to be done twice
- 10. Transformations “By changing the equation, you can change the shape, the direction it is pointing, and the location of the parabola”
- 11. General Parabola y= ax² By adding a negative sign before ‘a’ you can change the parabola from facing north to facing south y= -ax² x= ay² By adding a negative sign before ‘a’ you can change the parabola from facing east to facing west x= -ay²
- 12. Shape of Parabola y= 1x²y= 0.5x² y= 2x² In the equation y= ax², coefficient ‘a’ determines the shape of the parabola. Making it wider, or skinnier. If coefficient ‘a’ is greater than 0 but less than 1, then the parabola will increase in width making it look wider If coefficient ‘a’ is greater than 1, then the parabola will decrease in width making it look skinnier
- 13. Location of Parabola In the equation y= a (x-h) ² ‘h’ determines the horizontal movement of the parabola If h > 0 then the parabola shifts to the right If h < 0 then the parabola shifts to the left In the equation x= a (y-h) ² ‘h’ determines the vertical movement of the parabola If h < 0 then the parabola shifts to the upwards If h > 0 then the parabola shifts downwards y= a (x-2) ² x= a (y-2) ²
- 14. Location of Parabola (pt.2) In the equation y= a (x-h) ² + k ‘k’ determines the vertical movement of the parabola If k > 0 then the parabola shifts upwards If k < 0 then the parabola shifts downwards In the equation x= a (y-h) ² + k ‘k’ determines the horizontal movement of the parabola If k < 0 then the parabola shifts to the left If k > 0 then the parabola shifts to the right
- 15. Completing a Square y= ax²+bx+c Example One Example Two Example Three y= x² +6x-7 y= 3x² +24x+21 y= 2x² +5x+2 = (x² +6x+_9_)-7-9 = 3(x² +8x+_16_)+21-48 = 2(x² +5/2x+_25/16_)+2-25/8 = (x+3)²-16 = 3(x+4) ²-27 = 2(x+5/4)²-9/8 -to find the final term of any of these equations: divide 2nd term by 2 then square as shown - if there is a coefficient, factor it out as shown above - before putting in final term, you must multiply it by the coefficient - remember, what you do on one side, you must do to the other side 6/2 = 3 3² = 9 factor 3 out 8/2 = 4 4² = 16 factor 2 out 5/2 divide by 2 = 5/4 5/4² = 25/16
- 16. Word Problems Type One good idea to draw a picture of some sort A rectangular area is 600m. What are the dimensions of this area if there is a definite wall? A = lxw = (600-2w)(w) = 600w-2w² = -2w² +600w = -2(w²-300w+_22,500__)+45,000 = -2(w-150) ²+45, 000 W = 150 L = 600-2(150) = 300m W = 150m - remember, factor out coefficient first -divide second term by 2 then square to find the final term
- 17. Continued.. Type Two make a table to help you A company sells boots for $40, 600 people buy this product. For every $10 increase, 60 fewer people buy the boots. What is the maximum revenue? Price: 40+10x # Sold: 600-60x 40+10(3) = 70 600-60(3)= 420 (40+10x)(600-60x) = 2400-2400x+6000-600² = -600² + 3600x +2400 = -600(x²-6x + _9__)+24000 + 5400 = -600(x-3) ² +29 400 X = 3 http://www.sheepskin-boots-and- slippers.com/images/discount-ugg-boots.jpg
- 18. Continued.. Type Three the equation is always already given in the question A soccer ball is thrown up in the air with an initial velocity of 120m/s. Find the height of the ball and the time required with this equation: H = -5t² +120t + 4 h= -5t²+120t+4 = -5(t² -24t+ _144_)4+720 = -5(t-12) ² +724 t= 12 seconds h= 724 m http://www.albion.edu/imsports/