Graphing Quadratics

Step I. Graphing Quadratics from Standard form

Quadratics and their factorization.  Pitchfork terminology for quadratics. Dependence on parameters- Numerical Examples, Shifting and Scaling,

Each quadratic expression

y = ax²+bx+c

can be rewritten in the standard form

y = a[(x-h)²+k]

The latter is good form for graphing. How to rewrite is the subject of further lessons in steps II and III.

The points on the graph of

y = a[(x-h)²+k]

 can be obtained from by the following operation on the  graph of y=x² or its points.

1. A Horizontal Translation: Shift all points h = |h} units to the right if h is positive, or  shift all points |h| = -h units to the left if h is negative. Do nothing if h = 0.

2. A Vertical Translation: Then shift all points k = |k| units up if k is nonnegative, or  shift all points |k| = -k units down if h is negative. Do nothing if k = 0.

3. A Vertical Rescaling: Scale the y-coordinate) of each point on the graph by multiplying each ordinate (second coordinate) by the factor a.

The following numerical exercise should help your understand the shifts and vertical scaling.  At the end of the exercise, you may see how the first two may be combined into a single shift.

Numerical Exercise
= Numerical Experience

 Let h = 0.25, k = -1 and a = 0.5 = ½.in

y = a[(x-h)²+k]

First, create a table of values for y=x² for x in the interval [-2,2] but only those points for which x is a multiple of 0.5 (Well I have done the work for you, so you job is to double check the values - every detail matters.)

Observe the y-values are symmetric about x = 0. The minimum value in the table is at x = 0.

This first table of values leads to plot points

(-2,4), (-1.5, 2.25), (-1, 1), (-0.5, 0.25), (0,0),  (0.5,0.25), (1,1), (1.5, 2.25)  (2,4)

Exercise plot the points on graph paper where the horizontal x-scale includes the interval [-3,3] in divisions of 0.25, and the vertical  y-scale includes the interval [-1, 5] also in divisions or jumps of 0.25. Use solid dots for these points  Next join them by a smooth curve. to approximate the graph of  y=x²

Second. Apply the three steps above.

 Recall h = 0.25, k = 1 and a = 0.5 = ½.in

y = a[(x-h)²+k] = ½[(x-0.25)²+1]

Step 1: Plot x in the h-shifted interval

[-2+h, 2+h] = [-1.75, 2.25]

Observe the y-values are symmetric about x = 0.25 = h.. The minimum value in the table is at x = h.

This second table of values leads to plot points

(-1.75,4), (-1.25, 2.25), (-0.75, 1), (-0.25, 0.25), (0.25,0), (0.75,0.25), (1.25,1), (1.75, 2.25)  (2.25,4)

Notice that these points come from the first table points shifted to the right by 0.25 units.

Exercise plot the points on the same graph papers before. Use hollow circles for these points with a dot in the center.. Next join these dotted circles them by a smooth curve to approximate the graph of y=(x-½)²

Step 2: Recall  h = 0.25, k = 1 and a = 0.5 = ½.

 The previous step was independent of the value of k and a. This step depends on k = 1. We add  k = 1 to the ordinate, that is second coordinate or y coordinate, of each point.

This third table of values leads to plot points

(-1.75,5), (-1.25, 3.25), (-0.75, 2), (-0.25, 1.25), (0.25,1.0), (0.75,1.25), (1.25,2), (1.75, 3.25)  (2.25,5)

Notice that (i) these points come from the second table points shifted up by 1 units, and (ii) these same points also come from the first table points shifted up by 1 unit and to the right by 0.25 units, at the same, or with one shift followed by another.

Steps 1 and 2 could be combined together as a shift or translation not necessarily parallel to the coordinate axes.

Step 3: Recall h = 0.25, k = 1 and a a = 0.5 = ½.

The previous step(s) were independent of the value a. This step depends on the value a == 0.5 = ½. In it, we multiply each y-value of the previous step by the value of a to obtain the graph of

y= ½[(x-½)² +1] = a[(x-h)²+k]

Observe the y-values are still symmetric about x = h = 0.25 with a minimum at x = h = 0.25.

This fourth table of values leads to plot points

(-1.75,2.5), (-1.25, 1.625), (-0.75, 1), (-0.25, 0.625), (0.25,0.50), (0.75,0.625), (1.25,1), (1.75, 1.625)  (2.25,2.5)

The y-value, ordinate of these point come (i) from the third table y-values multiplied by a = 0.5;  or  (ii) the first table points shifted or translated by (h,k) = (0.25,1) followed by a multiplication of the y-value points by a = 0.5 = a vertical scale factor.

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