Quadratic Functions

I. Find the Zeros

Find the zeros of a given quadratic. The zeros are the x values when the curve crosses the x-axis (i.e. when y = 0)

Test Case:

Hints

II. Coordinates of the Vertex

Find the coordinates (x, y) of the vertex of the quadratic (it's maximum or minimum.)

III. Type of Critical Value (Max or Min)

Determine if the critical value is a maximum or minimum.

IV. Slope of the line.

Find the slope of a line tangent to the quadratic when: $$x = 2$$ If you have not taken calculus, see the hint to the right.

Hints

V. Chord Length

Given two x values (-2 and 2), find the length of a chord between the two points on the curve.

Hints

VI. Line Length

Estimate the length of the curve between x = -2 and x = 2 by using a series of chords along the line.

Hints

VII. Line Length: Taking it to the Limit

You can get a better/more accurate estimate of the distance along the curve by breaking the curve into smaller segments. Estimate the length of the curve between x = -2 and x = 2 by using more than just four line segments (use as many as you can).

We describe the process of adding up the sum of smaller and smaller cords as taking the limit as the change in x (dx) approaches zero; after all, if you made the step zero, you'd have infinite steps.

Mathematically, the limit as dx goes to zero is written as: $$\lim_{\Delta x \to 0}$$

Hints

VIII. Finding the Slope using the chord

You can also use the limit method to estimate the slope of the line.

Say, you want to find the slope when x = 2 (as in section IV). You could just take two points--when x = 1 and x = 2, and find the slope of the chord between those two points.

If we instead take the second point to be when x = 1.5 with the first point still x = 2, the difference in x (dx) is 0.5, and the slope becomes closer to the actual slope:

Mathematically, the slope at a point can be found by making dx as small as possible: i.e. taking the limit as dx goes to zero.

Find the slope at x = 2 by finding the slope of a very small chord.

Hints