# Quadratic Function Explorer Standard form

In standard form, a quadratic function is written as  y = ax2 + bx + c

In the applet below, move the sliders on the right to change the values of a, b and c and note the effects it has on the graph. See also Linear Explorer, Cubic Explorer.

See also General Function Explorer where you can graph up to three functions of your choice simultaneously using sliders for independent variables as above. See also Linear Explorer and Cubic Explorer.

## Things to try

Assuming you already have a knowledge of quadratic equations, the following activities can help you get a more intuitive feel for the action of the three coefficients a, b, c .

## The simplest case. Y = constant. (y = c)

1. Click 'zero' under the three sliders

Since a, b, c are all set to zero, this is the graph of the equation y = 0x2+0x+0. This simplifies to y = 0 and is of course zero for all values of x. Its graph is therefore a horizontal straight line through the origin.

1. Now move the rightmost slider for c and let it settle on, say, 12.

This is the graph of the equation y = 0x2+0x+12. This simplifies to y = 12 and so the function has the value 12 for all values of x. It is therefore a straight horizontal line through 12 on the y axis. Play with different values of c and observe the result.

## Linear equations. (y = bx)

1. Click 'zero' under the three sliders
2. Move the center slider to get different values of b. Let it settle on, say, 2.

This is the graph of the equation y = 0x2+2x+0 which simplifies to y = 2x. This is a simple linear equation and so is a straight line whose slope is 2. That is, y increases by 2 every time x increases by one. Since the slope is positive, the line slopes up and to the right. Adjust the b slider and observe the results, including negative values.

1. Now move both sliders b and c to some value.
This is the equation of y = bx+c and combines the effects of the b and c coefficients. Play with various values of b and c to get a feel for the effects of their values on the graph.

## The squared term. (y = ax2)

1. Click 'zero' under the three sliders
2. Move the left slider to get different values of a. Let it settle on, say, 3.

This is the graph of the equation y = 3x2+0x+0. This simplifies to y = 3x2. Equations of this form and are in the shape of a parabola, and since a is positive, it goes upwards on each side of the origin. Play with various values of a. As a gets larger the parabola gets steeper and 'narrower'. When a is negative it slopes downwards each side of the origin. Note also the roots of the equation (where y is zero) are at the origin and so are both zero.

## Full quadratic. (y = ax2+bx+c)

1. Click 'zero' under the three sliders
2. Set c to 4, the line moves up
3. Set b to 3, the line slopes.
4. Set a to 2. The parabola shape is added in.

This is the graph of the equation y = 2x2+3x+4. Note how it combines the effects of the three terms. Play with various values of a, b and c.

• Changing c moves it up and down.
• Changing b changes the slope.
• Changing a alters the curvature of the parabolic element.

## Roots

In the figure above, click on 'show roots'. As you play with the quadratic, note that the roots are where the curve intersects the x axis, where y = 0.   There are two roots since the curve intersects the x-axis twice, so there are two different values of x where y = 0. Under some circumstances the two roots may have the same value. If the curve does not intersect the x-axis at all, the quadratic has no real roots.

If you make b and c zero, you will see that both roots are in the same place. Under some conditions the curve never intersects the x-axis and so the equation has no real roots. Notice that if b = 0, then the roots are evenly spaced on each side of the origin, for example +2 and -2.  See "axis of symmetry" below.

When expressed in normal form, the roots of the quadratic are given by the formula below. It gives the location on the x-axis of the two roots and will only work if a is non-zero. If the expression inside the square root is negative, the curve does not intersect the x-axis and there are no real roots.

## Axis of symmetry

Click on "show axis of symmetry". This is a vertical line through the vertex of the curve. Note how the curve is a mirror image on the left and right of the line. (We say the curve is symmetrical about this line). Note too that the roots are equally spaced on each side of it.

When the quadratic is in normal form, as it is here, we can find the axis of symmetry from the formula below. It gives its location on the x-axis. If a is zero, there is no axis of symmetry and this formula will not work, the attempt to divide by zero will give an undefined result. In the figure above, set a to zero and moving the other sliders, convince yourself there can be no axis of symmetry with a = 0.

## Try it yourself

1. Press "reset", then "hide details"
2. Adjust the sliders until you see a shape that appeals to you
3. Estimate the values of a, b and c for this curve and write down the equation for the curve
4. Estimate the roots
5. Click on "show details" and see how close you got