Help appreciated~
1.Use the quadratic formula to determine when f(x)= x^2 +7x + 7 is greater than 0. Round your answer to two decimal places.
a) x<-5.79, x>-1.21
b) x<-7, x>1
c) 1<x<7
d) -5.79<x<-1.21
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2. On what interval will the function f(x) = x^2 - 5x -6 have a positive rate of change?
a) -1<x<6
b) x>2.5
c) x<-1, x>6
d) x is less than or equal to 2.5
1.
find the roots of the function: (-7 +/- sqrt(21))/2 = -5.79 and -1.21
Since the graph is a parabola which opens up, it will lie below the x-axis between the roots, and above the axis elsewhere.
A is the answer
2.
Again, you have a parabola which opens up. If you can find the vertex, f(x) will be decreasing on the left, and increasing on the right of the vertex.
The vertex, natch, is at x = -b/2a = 5/2.
So, B is the answer
1. Oh, quadratic formulas, such mathematical wonders! Time to put on my thinking cap... or should I say, my clown wig? Ready? Here we go! *honk honk*
To determine when f(x) = x^2 + 7x + 7 is greater than 0, we need to find the x-values that make the function positive. We can solve this using the quadratic formula (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 7, and c = 7.
Calculating... drumroll, please... *drum roll* The answer is option a) x < -5.79, x > -1.21! So, if your x-values fall within these ranges, you'll have some positive fun with this function. Clowns love positive fun!
2. Ah, rate of change, a fancy way of saying "how things are moving." Let's see what we can do here. *rubs clown nose*
For f(x) = x^2 - 5x - 6 to have a positive rate of change, we need to find the interval where the function is increasing. To do this, we can find the x-values where the derivative of the quadratic function is positive.
Calculating... spinning plates... juggling balls... *clown magic* The answer is option c) x < -1, x > 6! So, if your x-values are either less than -1 or greater than 6, you'll be in the land of positive rate of change. Break out the clown shoes and start dancing!
1. To determine when f(x) = x^2 + 7x + 7 is greater than 0, we can use the quadratic formula. The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x can be found using:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 7, and c = 7. Plugging these values into the quadratic formula, we have:
x = (-(7) ± √((7)^2 - 4(1)(7))) / (2(1))
x = (-7 ± √(49 - 28)) / 2
x = (-7 ± √(21)) / 2
x = (-7 ± 4.58) / 2
Simplifying further, we get:
x = (-7 + 4.58) / 2 = -1.21
x = (-7 - 4.58) / 2 = -5.79
Therefore, the solutions for x are -1.21 and -5.79.
To determine when f(x) > 0, we are looking for the values of x that make f(x) positive. This occurs when x is greater than -5.79 and less than -1.21.
So the answer is: d) -5.79 < x < -1.21.
2. To find the interval where the function f(x) = x^2 - 5x - 6 has a positive rate of change, we need to find when the derivative of f(x) is positive. The derivative of f(x) can be found by taking the derivative of each term separately:
f'(x) = 2x - 5
For the function to have a positive rate of change, we want f'(x) > 0. Solving the inequality, we have:
2x - 5 > 0
2x > 5
x > 5/2
Therefore, the function f(x) = x^2 - 5x - 6 has a positive rate of change when x is greater than 5/2.
So the answer is: b) x > 2.5.
1. To determine when the quadratic function f(x) = x^2 + 7x + 7 is greater than 0, we can use the quadratic formula.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation, ax^2 + bx + c = 0.
In this case, a = 1, b = 7, and c = 7. Plugging these values into the quadratic formula, we get:
x = (-7 ± √(7^2 - 4(1)(7)))/(2*1)
x = (-7 ± √(49 - 28))/(2)
x = (-7 ± √(21))/(2)
x ≈ (-7 ± 4.58)/(2)
Rounding the solutions to two decimal places, we have x ≈ -5.79 and x ≈ -1.21.
So, the answer is option a): x < -5.79, x > -1.21.
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2. To find the interval where the function f(x) = x^2 - 5x - 6 has a positive rate of change, we need to find the values of x for which the derivative of the function is positive.
The rate of change of a function is given by its derivative. In this case, the derivative of f(x) = x^2 - 5x - 6 is f'(x) = 2x - 5.
To find when f'(x) > 0, we set 2x - 5 > 0 and solve for x:
2x - 5 > 0
2x > 5
x > 5/2
x > 2.5
So, the answer is option b): x > 2.5.