Find the smallest positive angle between the lines with equations y=3/2x+8 and y=1/5x-6.
Let's make Θ1 the angle between the first equation and the positive x-axis. Then let's make Θ2 the angle between the second equation and positive x-axis. This would mean that the tan(Θ) =tan (Θ1-Θ2) We know that the tan (Θ1) = slope of the first equation which is 3/2 and the tan (Θ2) = slope of the second equation with is 1/5. Based on our identities we know that tan (Θ1 -Θ2)= (tan(Θ1)-tan(Θ2))/(1+tan (Θ1) * tan(Θ2)) After we plug in the numbers, we get that the tan (Θ1 -Θ2)=1 so the angle between the two lines is 45 degrees
Integrating just means reversing a derivative. To integrate a x^n term, take the exponent and add one to it to make the new exponent. Then take the new exponent value and divide the term by it So for this problem 6x^5 would be x^6. You can check the answer by taking the derivative. Bring down the 6 and make it the coefficient in front and subtract the exponent by 1. This would make it 6x^5 so we know our answer is correct. If we did the whole problem it would be x^6 +6x^3+7x +c. Don't forget the +c term since it is an indefinite integral. The original function may have had a constant so it's important to add the +c term.
Given the problem: ax^2 + bx + c = 0, find the solutions and graph the parabola
First, the solutions can be found using the quadratic formula --> x=(-b ± sqrt(b^2-4*a*c))/(2*a) By substituting the coefficients a, b, and c you can obtain the solutions The discriminant, the part that is under the square root symbol (b^2-4*a*c) will tell how many solutions there are. If the discriminant is greater than zero, then there will be two different, real solutions. If the discriminant is equal to zero, then there will be one repeating real solution. If the discriminant is less than zero, there-there will be two nonreal solutions meaning two complex solutions involving the imaginary number i. If the coefficient a is positive, then the parabola will open upwards If the coefficient a is negative, then the parabola will open downwards The y-intercept is found by making x=0 and solving for the y value that makes it true The x-intercept is found using the quadratic formula above. There can be 0, 1, or 2 x-intercepts. The x-coordinate of a vertex is found by using the formula x=-b/2a. To find the y-coordinate, plug the x-value into the original equation. Plot the vertex, x-intercept(s), and y-intercepts. Then create a curved line connecting them to form the parabola