Ask Dr. Math

High School Archive

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

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

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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 [07/13/1999]

Solve the inequality and express the solution in terms of intervals.

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?

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)...

Translation [9/11/1996]

What does translation mean?

TV Transmitters [07/22/1997]

If two TV transmitters are within 150 kilometers of each other, they must be assigned different channels. Construct a graph that shows the given problem and use it to find the number of channels needed.

Types of Linear Systems [10/13/1997]

What is meant when a type of linear system is said to be a consistent or an inconsistent system?

Understanding Parametric Equations [04/02/1998]

Can you explain the basic principles of parametric equations?

Units for Inverse Quantity [9/10/1996]

I have a graph of time vs. 1/distance. What units do I use for 1/ distance - cm^-1?

Uses for Parametric Equations [04/23/1998]

What is a parametric equation used for? Can you give me an example?

Using Cardan's Formula to find Real Roots [04/13/2000]

How can I show that, although the cubic equation x^3 - 6x = 4 has three real solutions, Cardan's formula can find them by subtracting appropriate cube roots of complex numbers?

Using Descartes' Method of Tangents [10/2/1995]

Given a point P(3,9) on a parabola y=x^2, how do I find the slope of the tangent to the parabola at point P using Descartes' method of tangents?

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