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

**a) y=-3x ^{2}**

**b) y=1/4 x ^{2}**

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=-3x ^{2}** 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 x ^{2}**

**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):