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Browse High School Equations, Graphs, Translations
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Direct and indirect variation.
    Slope, slope-intercept, standard form.

Polar Coordinates and Logarithmic Spirals [04/25/2004]
The polar coordinate graph of a logarithmic spiral suggests that for a given angle, there is more than one possible r-value, since the graph continues to spiral around itself. How is it possible for an angle to be associated with more than one r?

Polar Coordinates From Cartesian Coordinates [07/27/1998]
How do you find the polar coordinates from the Cartesian coordinate (3, - 3 sqrt(3))?

Polar Equation of a Circle [4/12/1997]
Given the polar equations r = a*sin (theta) and r = a*cos (theta), what influence does a have on the graph?

Polar to Rectangular Conversions [4/19/1996]
How do I convert "8 cis 30" into rectangular coordinates?

Polynomial Function and Exponential Growth [06/08/1999]
How can I find a polynomial function with integer coefficients that has 3, 2, -4 and -1 as zeroes? What is the population of a colony of bacteria 20 seconds after it is discovered?

Practical Uses for Computing Parabolas [1/30/1996]
Could you give me three practical uses for computing parabolas and tell me the significance of computing the parabola in regard to its application?

Pre-calculus Function Question [2/8/1996]
Does the function f(x)=3De^(1/x) have a local minimum?

Proof of Perpendicularity [10/23/1999]
How can you prove that two lines (neither vertical) are perpendicular if and only if the product of the gradients is equal to -1?

Quadrant [03/25/2001]
What is a quadrant?

Quadratic Formulas, Equations, Parabolas, Graphing [6/16/1996]
Given 6x^2 + x - 1 = 0, how do I find the roots, the vertex, some coordinates, and from these graph it?

Quadratic Inequalities [03/18/2003]
When I solve (x^2 + 1)/4 greater than or equal to (x + 2)/2 , of my final two answers, only one works.

Quintic Equations [1/27/1996]
What does Quintic mean?

Recognizing Functions [09/30/2001]
How do I know when there is a function on a graph?

Rectangular Hyperbola [06/11/2002]
The graph of a rectangular hyperbola looks nothing like a rectangle. Where does the name come from?

Root Multiplicity and Polynomial Functions [11/16/1997]
What effect does multiplicity [e.g. (x+1)(x-2)^2 where -1 has a multiplicity of 1 and 2 of 2] have on a polynomial function?

The Roses of Grandi [11/13/2000]
The Roses of Grandi are given by polar equations r = a*sin(n*theta) and r = a*cos(n*theta) where n is an integer. How can I show that they are algebraic?

Rotating Parabolas [01/19/1999]
How do you find a parabola that has a line of symmetry other than a line parallel to the x-axis or y-axis?

Shape of the Sine Curve vs. Circle [07/18/2001]
When the equation y = 5sinx is graphed, the curve looks almost linear except near the points where y = 5. Since the sine curve originates from the circle, why isn't it completely curvy like a circle?

Shifting Graphs [08/26/2002]
Why is the graph of y=f(x-c) just the graph of y=f(x) shifted c units to the right?

Similar Graphs [12/11/1995]
How would you describe how the two graphs of y = one over the square root of x squared and y = one over the square root of x squared minus 4 are similar and dissimilar?

Sketch a Graph [6/1/1996]
z=(3-2i)^1/2, then find z^.

Sketching a Function [07/06/1998]
Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ...

Sketching a Polynomial [04/04/2002]
Why should a curve change its position or sign at roots? Can't a curve have a positive value for all roots?

Sketching Curves [6/24/1996]
How do I sketch the curve y = log natural of x/ (1-x)?

Slope-Intercept Formula [06/05/2001]
Why does y = mx + b have to be that way?

Slope of Lines [8/6/1996]
What is the slope of the line y = -px + 5 if the line goes through (0, - 3)? Why does the y-axis have an infinite slope, and the x-axis have a slope of 0?

Slopes of Perpendicular Lines [09/17/1998]
Why are the slopes of perpendicular lines negative reciprocals?

Solve for Radius [09/09/1997]
Given a circle through three points, what is the equation for the intersection of the perpendicular bisectors?

Solving a Fractional Equation [3/10/1996]
Can you help me with this? (M+N)/(N-M) + (M^2+N^2)/(N^2-M^2) - (2M)/(M-N) = ?

Solving Conic Equations [05/31/1999]
Can you show me the steps for solving conic equations and putting them in hyperbola, ellipse, or parabola form?

Solving Inequalities: Cubic Curve, and Case by Case [07/13/1999]
Solve the inequality and express the solution in terms of intervals.

Some Doubt about Absolute Value ... in Three Parts? [10/03/2016]
A teen wonders how to directly plot an absolute value that appears to have more than two distinct intervals. Thanks to the function's continuity, Doctor Peterson picks up where she left off to re-write the pieces into something easier to visualize.

Standard Form in Linear Equations [11/21/2001]
Where is standard form used?

The Sum of a Number and Its Reciprocal [11/04/2003]
I need a formal proof showing that the sum of a positive number and its reciprocal is at least 2. I can prove it algebraically, but I need a visual justification.

Symmetry Tests [01/12/1999]
How can you know whether a graph is symmetric to the x-axis, y-axis, or the origin? What does the symmetry mean?

Tangents to a Curve That Pass through a Fixed Point [08/08/2008]
How many tangent lines to the curve y = x/(x+1) pass through the point (1,2)? At which points do those tangent lines touch the curve?

T-Charts and Graphs [11/11/2001]
Make a table of values for the line y = x/4+4 using x-values of 1,2,3,4, and 5. Which graph shows the line that passes through the ordered pairs of the table?

Testing for Horizontal Asymptotes [09/12/1998]
Is there a rule for testing whether or not an equation has a horizontal asymptote?

Three Points, Two Unknown x-coordinates, All in a Row? [03/04/2013]
Given three points, two with unknown abscissas, a student has doubts about their collinearity. After offering some leading hints, Doctor Mike invokes the point-slope form to rigorously confirm the student's suspicions.

Translating Functions [08/27/1998]
Find f(2+h), f(x+h), and f(x+h)-f(x)/h where h cannot = 0 for f(x) = x/(x + 1). Explain how the following graphs are obtained from the graph of y = f(x)...

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