If the pattern rule is x2 +6, imagine what the trend line for this pattern would look like (on a graph of course). What pattern rule would have a trend line at the same vertical intercept (y-intercept) but has a different steepness? Explain your thinking. PLEASE HELP ME.

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y = x^2 + 6 looks like this:

http://www.wolframalpha.com/input/?i=y+%3D+x%5E2+%2B+6

now let's look at
y = 3x^2 + 6 and y = (4/5)x^2 + 6 on the same grid
http://www.wolframalpha.com/input/?i=y+%3D+3x%5E2+%2B+6%2C+y%3D%284%2F5%29x%5E2+%2B+6

what do you think?

What pattern rule would have a trend line that starts at the same vertical intercept (y-intercept) but has a different steepness?

To understand the trend line for the given pattern rule, let's plot some points on a graph to see the shape of the line it forms.

For the pattern rule x^2 + 6, we can choose some values of x and compute the corresponding y values to form points:

When x = -3, y = (-3)^2 + 6 = 15
When x = -2, y = (-2)^2 + 6 = 10
When x = -1, y = (-1)^2 + 6 = 7
When x = 0, y = (0)^2 + 6 = 6
When x = 1, y = (1)^2 + 6 = 7
When x = 2, y = (2)^2 + 6 = 10
When x = 3, y = (3)^2 + 6 = 15

Plotting these points on a graph, we get a U-shaped curve that opens upwards. The line passes through the point (0, 6), which represents the y-intercept (vertical intercept).

Now, to find a pattern rule with the same y-intercept but a different steepness, we need to consider a pattern that will have a similar shape but with a different rate of change.

One pattern that fits this description is a linear pattern. A linear pattern has a constant rate of change, meaning it forms a straight line on the graph.

Let's consider the linear pattern y = 2x + 6. This pattern has the same y-intercept (vertical intercept) of 6, but the steepness is determined by the coefficient of x, which is 2 in this case. The coefficient of x determines the slope or steepness of the line.

If we plot some points for this linear pattern:

When x = -3, y = 2(-3) + 6 = 0
When x = -2, y = 2(-2) + 6 = 2
When x = -1, y = 2(-1) + 6 = 4
When x = 0, y = 2(0) + 6 = 6
When x = 1, y = 2(1) + 6 = 8
When x = 2, y = 2(2) + 6 = 10
When x = 3, y = 2(3) + 6 = 12

Plotting these points on a graph, we get a straight line with the same y-intercept (0, 6). The steepness of this line is different from the quadratic pattern since it forms a straight line rather than a curve.

In summary, the pattern rule x^2 + 6 creates a U-shaped curve with a vertical intercept of (0, 6). To find a different pattern rule with the same y-intercept but a different steepness, we can use a linear pattern, such as y = 2x + 6, which forms a straight line.