What is the rate of change in y with respect to x in the function x + 5y = 10?
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Answer:
Step-by-step explanation:
Part A:
To complete the square, we will do the first few steps all at once, since they are necessary but not complicated to follow all at once. Set the function equal to 0 first, then move the constant over to the other side:
The most important rule for completing the square is that the leading coefficient (the number on the t-squared term) is a positive 1. Ours is, but if it wasn't it would need to be factored out.
Next, take half the linear term, square it, and add it to both sides. Our linear term is 6. Half of 6 is 3, and 3 squared is 9, so we add 9 to both sides:
On the right is simple addition, but on the left side, during this process we created a perfect square binomial that will reflect the h coordinate of the vertex. We will state that binomial and do the addition on the right both at the same time:
If you FOIL out the (t + 3)(t + 3) you will get back the polynomial
Now we can move the constant back over and set the polynomial back to equal f(t):
From this we determine the vertex to sit at (-3, -38)
Part B:
Because this is a positive parabola that opens upwards, the function will have a minimum value (if it was negative and opened upside down, it would have a minimum value). In other words, you know the vertex is a minimum because this is a positive parabola that opens upwards.
Part C:
The axis of symmetry is the line that cuts the parabola in half, creating 2 identical, mirror images. Because this parabola opens upwards, a vertical line is required to cut it into 2 identical halves. The vertical line has an equation " x = ". In the case of a positive parabola that opens upwards, the line of symmetry, what x = , will ALWAYS be the number that is inside the parenthesis with the x. Since the standard vertex form is
our vertex can be rewritten as (t - (-3)). Therefore, our axis of symmetry is
x = -3