Find an equation for the line below. Thanks !
2 answers:
Answer:
Step-by-step explanation:
In order to find the equation to this line, we need to note that we see two points on this graph. Using these two points, we can use them to find the slope of the graph and use one to find the y-intercept.
We know that the slope of a line is defined by (change in y / change in x). Therefore, we can use our two points that we know - (-5, 2) and (3, 4) to find the slope.
The change in y is , and the change in x is
. Therefore, our slope is
.
Now that we know our slope, our equation in slope-intercept form looks something like this.
However, we still have b to solve for. We can solve for this by substituting a point we already know into the equation. Let's substitute (3, 4) inside.
So now we know that the y-intercept is . Plugging that into our equation finishes it off, leaving our final equation to be
.
Hope this helped!
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Answers:
(a) f is increasing at
.
(b) f is decreasing at
.
(c) f is concave up at
(d) f is concave down at
Explanations:
(a) f is increasing when the derivative is positive. So, we find values of x such that the derivative is positive. Note that
So,
The zeroes of (x - 2)(x + 2) are 2 and -2. So we can obtain sign of (x - 2)(x + 2) by considering the following possible values of x:
-->> x < -2
-->> -2 < x < 2
--->> x > 2
If x < -2, then (x - 2) and (x + 2) are both negative. Thus, (x - 2)(x + 2) > 0.
If -2 < x < 2, then x + 2 is positive but x - 2 is negative. So, (x - 2)(x + 2) < 0.
If x > 2, then (x - 2) and (x + 2) are both positive. Thus, (x - 2)(x + 2) > 0.
So, (x - 2)(x + 2) is positive when x < -2 or x > 2. Since
Thus, f'(x) > 0 only when x < -2 or x > 2. Hence f is increasing at.
(b) f is decreasing only when the derivative of f is negative. Since
Using the similar computation in (a),
Based on the computation in (a), (x - 2)(x + 2) < 0 only when -2 < x < 2.
Thus, f'(x) < 0 if and only if -2 < x < 2. Hence f is decreasing at (-2, 2)
(c) f is concave up if and only if the second derivative of f is positive. Note that
Since,
Therefore, f is concave up at.
(d) Note that f is concave down if and only if the second derivative of f is negative. Since,
Using the similar computation in (c),
Therefore, f is concave down at.
(a) f is increasing at
(b) f is decreasing at
(c) f is concave up at
(d) f is concave down at
Explanations:
(a) f is increasing when the derivative is positive. So, we find values of x such that the derivative is positive. Note that
So,
The zeroes of (x - 2)(x + 2) are 2 and -2. So we can obtain sign of (x - 2)(x + 2) by considering the following possible values of x:
-->> x < -2
-->> -2 < x < 2
--->> x > 2
If x < -2, then (x - 2) and (x + 2) are both negative. Thus, (x - 2)(x + 2) > 0.
If -2 < x < 2, then x + 2 is positive but x - 2 is negative. So, (x - 2)(x + 2) < 0.
If x > 2, then (x - 2) and (x + 2) are both positive. Thus, (x - 2)(x + 2) > 0.
So, (x - 2)(x + 2) is positive when x < -2 or x > 2. Since
Thus, f'(x) > 0 only when x < -2 or x > 2. Hence f is increasing at
(b) f is decreasing only when the derivative of f is negative. Since
Using the similar computation in (a),
Based on the computation in (a), (x - 2)(x + 2) < 0 only when -2 < x < 2.
Thus, f'(x) < 0 if and only if -2 < x < 2. Hence f is decreasing at (-2, 2)
(c) f is concave up if and only if the second derivative of f is positive. Note that
Since,
Therefore, f is concave up at
(d) Note that f is concave down if and only if the second derivative of f is negative. Since,
Using the similar computation in (c),
Therefore, f is concave down at