y = 3(x + 4)^2 + 31
Step-by-step explanation:
We can convert the given quadratic equation into its vertex form by completing the square:
y = 3x^2 + 24x + 43
= 3(x^2 + 8x) + 43
= 3(x^2 + 8x + 4) + 31
= 3(x + 4)^2 + 31
This is the vertex form of the given quadratic equation with (-4, 31) as its vertex
Answer: 3ln4 + 3lna
Step-by-step explanation:
Answers:
The formula is [f(-1)-f(-4)]/[3]
The value of f(-1) is 3
The value of f(-4) is -3
The average rate of change is 2
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Explanation:
For the first blank, we use the formula
[ f(b) - f(a) ]/[ b - a ]
where 'a' and 'b' are the endpoints for the x interval
In this case, a = -4 and b = -1. When you plug those values into the formula above, you get...
[ f(b) - f(a) ]/[ b - a]
[ f(-1) - f(-4)]/[ -1 - (-4) ]
[ f(-1) - f(-4)]/[ -1+4 ]
[ f(-1) - f(-4)]/[ 3 ]
which is why the answer is choice C for the first blank
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To compute the value of f(-1), we draw a vertical line through -1 on the x axis. This vertical line crosses the diagonal function graph at the point (-1,3). The y value of this point is what we want. Plugging in x = -1 leads to y = 3. This is why f(-1) = 3
If you want, you can draw a horizontal line through (-1,3) and you'll see it touching 3 on the y axis.
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Follow similar steps as above to compute f(-4). Draw a vertical line through x = -4 on the x axis. Mark the point where the vertical line crosses the diagonal line. This point is (-4,-3). Optionally draw a horizontal line over til you hit the y axis and you'll find that y = -3 corresponds to x = -4
This is why f(-4) = -3
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We'll use the last three sections to compute the average rate of change. Everything combines together building up to this moment.
From the first part, we had the formula
[ f(b) - f(a) ]/[ b - a ]
[ f(-1) - f(-4)]/[ 3 ]
We can replace the "f(-1)" with 3 since we found that f(-1) = 3
Similarly, f(-4) = -3 so we can replace the "f(-4)" with -3
Doing those replacements and simplifying leads to...
[ f(-1) - f(-4)]/[ 3 ]
[ 3 - (-3)]/[ 3 ]
[ 3 + 3]/[ 3 ]
6/3
2
So the average rate of change is 2
Note: because the entire graph is a straight line, the average rate of change for any interval a < x < b is going to be equal to the slope m. In this case, the slope of the line is m = 2/1 = 2. We move up 2 units each time we move to the right 1 unit along the diagonal line.
Answer:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45
Step-by-step explanation:
B I think but wait for other people to answer
