Why is multiplying a number by 4/4 or 2/2 the same as multiplying it by 1?
4 divided by 4 equals 1, and so does 2 divided by 2. So, when you multiply the number by 2 and then divide it by 2, it is the same as if you had just multiplied it by one. Another way to think of it is in terms of fractions and whole numbers. When the numerator and denominator in a fraction are equivalent, or equal to each other, then the fraction is equal to one. For example, if a pizza is cut into 8 slices, and you still have 8 slices, then you still have the full pizza. But, as soon as you add or take away a slice, you no longer have a full pizza, so the fraction no longer equals one.
Using calculus terms, describe the relationship between velocity and acceleration.
The derivative of velocity is acceleration. This is because acceleration describes the rate at which velocity is changing, and derivatives describe rates of change. Derivatives are similar to slope, which also describe the rate of change of a curve or line. On the other hand, the integral of acceleration is velocity, because an integral is an antiderivative. According to the Fundamental Theorem of Calculus, the antiderivative of a derivative is the function itself, which is why velocity is retrieved when the acceleration is integrated.
At which coordinate point does the quadratic equation y=1-x^2 intersect the y axis? What about the x axis?
Y-axis: To find this point, you set x=0, because the y-axis is the collection of points on the xy-plane in which x=0. So, y=1-0^2, and y=1. In coordinate form, (0,1). X-axis: Because we know that the intersection on the y-axis is above the origin and the curve opens down, we know that there has to be two different intersection points on the x-axis. Similarly to finding the y-axis intersection, we set y=0 this time. So, 0=1-x^2, we get that x=1 and x=-1. This is because once x is squared, the negative is eliminated, so both roots make the equation true. In coordinate form, (-1,0) and (1,0).