Question

Without Graphing, describe each function as it would compare to y=x^2. 1) y=2x^2+100 2) y=1/20x^2-36 Describe the effects of changing "a" on the graph of quadratic...

Answer 1

The functions are y = x^2, 1).y = 2x^2 + 100, and 2).y = (1/20)x^2 - 36.

Note :

Every function listed above has something in common : they alll contain an x^2 as its highest power.

This means that all of these functions are quadratic functions, and all will form a parabola when graphed.

The only differences will be in the direction of opening(up or down), the size (compressed or stretched), and/or the location of the vertex.

In the function y= x^2, x^2 coeffient is positive 1, so the graph opens upward.

1). y = 2x^2 + 100

For a quadratic equation, the vertex is the locator point.

The vertex for the quadratic equation f(x) = a(x - h)^2 + K is the point (h, k).

Rewrite the equation as y = 2(x - 0)^2 + 100.

So, the vertex is (0, 100) and a = 2.

The  a is 2 causes , the graph to be skinneier and open upward.

The 0 causes, it to don ' t shift to the any units, and +100 causes the graph to shift up 100 units.

2). y = (1/20)x^2 - 36.

the vertex for the quadratic equation f(x) = a(x - h)^2 + K is the point (h, k).

Rewrite the equation as y = (1 / 20)(x - 0)^2 - 36.

So, the vertex is (0, - 36) and a = 1 / 20.

The  1/20 will compress the parabola, making it appear "fatter".

The 0 causes, it to don ' t shift to the any units, and -36 causes the graph to shift down 36 units.

All these graphs are shown in below.

Answer 2

The nonrigid transformation of the graphs of y = f(x) is represented by g(x) = c f(x),  where c > 1 then transformation is vertical stretch, where 0 < c < 1 then transformation is vertical shrink.

The Vertical and horizontal shifts in the graphs of y = f(x) are represented as follows:

1. Vertical shift c units upward: h(x ) = f(x ) + c.

2. Vertical shift c units downward: h(x ) = f(x ) - c.

Let the parent function is f(x ) = x² and the transformation functions are h(x ) = 2x² + 100 and g(x ) = (1/20)x² - 36.

The graph of h(x ) = 2x² + 100 = 2 f(x ) + 100 is a vertical stretch (c = 2 > 1) followed by an upward shift of 100 units of the graph of f(x).

Therefore each y - value is multiplied by 2 and add 100.

The graph of g(x) = (1/20)x² - 36 = (1/20) f(x) - 36 is a vertical shrink (0 < c = 1/20 = 0.05 < 1) followed by an downward shift of 36 units of the graph of f(x).

Therefore each y - value is multiplied by 1/20 and subtract 36.