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Understanding Functions: Domain, Range, Intercepts, and Graphical Transformations in AP Calculus

This guide breaks down key concepts in AP Calculus, focusing on functions' domains and ranges. The domain identifies all x-values for which a function is defined, while the range encompasses all y-values. Learn to find intercepts by solving equations, and explore intersections of graphs algebraically through substitution or elimination. The guide also highlights the effects of transformations on parent graphs and the determination of vertical asymptotes. Use provided calculator tips for verifying intersections and understanding graph behavior.

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Understanding Functions: Domain, Range, Intercepts, and Graphical Transformations in AP Calculus

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  1. Analyzing Graphs AP Calculus AB Functions

  2. Domain & Range Domain: All x values for which a function is defined. All real numbers for which the equation is defined. implicit (i.e. implied) explicit (i.e. defined) Given with the function. Range: All y values for which a function is defined. y is the dependent variable – Its value depends on x.

  3. Interval Notation

  4. Intercepts The intercepts of a graph are the points at which it intersects an axis. x-intercepts: Let y = 0 and solve. y-intercepts: Let x = 0 and solve. Given , find all intercepts.

  5. Intersections When solving systems of 2 or more equations, the solution(s) are the intersections of the graphs. To solve algebraically use substitution or elimination. Find all points of intersection of . CALCULATOR TIP: Use the calculator to verify the solutions. -Enter the functions into Y=. -Go to CALC (2nd Trace). -Choose INTERSECT (5). -Move the cursor near the intersection you want it to find.

  6. Transformations When a parent graph has been transformed, its function is altered in the following ways: h (x)  transformed function f (x)  parent function

  7. Inverses Graphically, the inverse of a function is its reflection over the line y = x. The coordinates of each point of the original function interchange to achieve a point of the inverse function.

  8. Vertical Asymptotes To find the vertical asymptotes of a function, set the denominator equal to zero and find all x-values for which the function is undefined. Find the vertical asymptotes of .

  9. Other Discontinuities Some graphs have horizontal asymptotes instead of or in addition to vertical asymptotes.

  10. Other Discontinuities Removable discontinuities are single points where there are holes in the graph.

  11. Assignment p. 8: 1-4, 15-20, 61-64 p. 27: 1, 2, 13, 14, 17, 18, 47-53, 55 p. 347: 6-12

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