Graphing Quadratics

Q:  Sketch graphs of these three quadratics relations on the same set of axes

a) y=-3x2

b) y=1/4 x2
A:  There are many ways to sketch graphs (quadratics).  The most basic way is to:  (1) make a table of values, (2) plot the points, (3) connect the dots.

So, let’s start with the first equation (a) y=-3x2 and make a table of values:

STEP 1

Start by picking values for x.  I will pick -2, -1, 0, 1, 2 for x:

x  | -2 | -1 | 0 | 1 | 2

y  |     |     |      |    |

Now we need to calculate the y values.

When x = -2, y=-3(-2)2 = -3(4) = -12

When x = -1, y=-3(-1)2 = -3(1) = -3

When x = 0, y=-3(0)2 = -3(0) = 0

When x = 1, y=-3(1)2 = -3(1) = -3

When x = 2, y=-3(2)2 = -3(4) = -12

So, the table is complete:

x  | -2   |  -1  | 0   | 1   | 2

y  | -12 |  -3 |  0  | -3 |  -12

STEP 2:

Plot the points: (-2, -12)  (-1, -3)  (0, 0)  (1, -3)  (2, -12)  on the axes

STEP 3

Connect the dots to make a parabola.  You should have a picture like so:

Now we need to do the same thing for equation (b) y=1/4 x2

STEP 1

Start by picking values for x.  I will pick -2, -1, 0, 1, 2 for x:

x  | -2 | -1 | 0 | 1 | 2

y  |     |     |      |    |

Now we need to calculate the y values.

When x = -2, y=1/4 (-2)2 = 1/4 (4) = 1

When x = -1, y=1/4 (-1)2 = 1/4 (1) = 1/4

When x = 0, y=1/4 (0)2 = 1/4 (0) = 0

When x = 1, y=1/4 (1)2 = 1/4 (1) = 1/4

When x = 2, y=1/4 (2)2 = 1/4 (4) = 1

So, the table is complete:

x  | -2   |  -1    | 0   |   1   | 2

y  |  1    |  1/4 |  0  | 1/4 |  1

STEP 2:

Plot the points: (-2, 1)  (-1, 1/4)  (0, 0)  (1, 1/4)  (2, 1)  on the axes

STEP 3

Connect the dots to make a parabola.  You should have a picture like so:

SO…. If we wanted to put these two graphs on the same axes, it would look something like (depending on your scale on the y-axis):

Solve for x (complicated)

Q:  Solve for x:

-2[8 – 5(2-3x) – 7x] = 4(x – |-9|)

A:

There are many ways to start with this, so I’m just going to start on the right side of the equation first:

-2[8 – 5(2-3x) – 7x] = 4(x – |-9|)

Starting on the right (piece at a time)

We know that |-9| = 9 (by definition of absolute value), so:

4(x – |-9|)

4(x – 9)

Now use the distributive property to distribute the 4 through (multiply the 4 to the x and the 9):

4x – 36

This is the best we can do on the right side.  So the right side (for now) is 4x – 36.

Now let’s look at the left side:

-2[8 – 5(2 – 3x) – 7x]

We need to get rid of the inner most parentheses, so we should deal with the -5(2 – 3x) part.  Distribute the -5 through:

-2[8 – 10 + 15x – 7x] <– that is what happens on the left when the -5 distributed through.

Now, clean up inside the brackets and combine like terms:

-2[-2 +8x]  <— I combined the 8-10 and the 15x-7x

Now distribute the -2 through the brackets to get:

4 – 16x  <– this is as far as the left side can be simplified.  So, combining the left side = right side we get:

4 – 16x = 4x – 36

I’m going to add 16x to both sides (to get rid of the x on the left side):

4 = 20x – 36

Now add 36 to both sides:

40 = 20x

Divide both sides by 20 to get x by itself:

2 = x [final answer]