Let p(x,y) be a point on the graph of y= X^2-3
A) Express the distance from P to the point (1,2) as a function of x. Simplify completely.
B) Use your calculator to determine which value of x-yields the smallest d.
Square root of x^4-5x^2+9
d = √[(x-1)^2 + (y-2)^2]
= √[(x-1)^2 + (x^2-3-2)^2]
= √(x^2-2x+1 + x^4-10x^2+25)
= √(x^4-9x^2-2x+26)
A) To find the distance between two points (x1, y1) and (x2, y2), we can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the point (1, 2) corresponds to (x1, y1) and the point on the graph y = x^2 - 3 corresponds to (x, y), which is (x, x^2 - 3).
Substituting these values into the distance formula, we get:
d = √((x - 1)^2 + (x^2 - 3 - 2)^2)
= √((x - 1)^2 + (x^2 - 5)^2)
Simplifying this expression further is not possible because it involves a sum of squares.
B) To determine which value of x yields the smallest d, we need to find the minimum value of the expression inside the square root (√(...)).
One way to do this is by using a graphing calculator. By plotting the function y = √((x - 1)^2 + (x^2 - 5)^2) and using the trace or minimum function, we can find the x-value that corresponds to the minimum value of d.
Alternatively, we can solve for the derivative of the expression and find the critical points. However, the resulting equation is quite complicated, and it would be easier and more accurate to use a calculator to find the minimum value.
To express the distance from point P on the graph of y = x^2 - 3 to the point (1, 2) as a function of x, we can use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values of (x1, y1) = (1, 2) and (x2, y2) = (x, x^2 - 3) into the distance formula, we have:
d = √((x - 1)^2 + (x^2 - 3 - 2)^2)
Simplifying this expression, we get:
d = √((x - 1)^2 + (x^2 - 5)^2)
Therefore, the distance from point P to the point (1, 2) as a function of x is given by:
f(x) = √((x - 1)^2 + (x^2 - 5)^2)
To determine which value of x yields the smallest distance, we can use a calculator to find the minimum value of the function f(x).
Here are the steps to find the value of x that yields the smallest distance:
1. Enter the function f(x) = √((x - 1)^2 + (x^2 - 5)^2) into your calculator or a graphing tool.
2. Use the calculator's built-in minimum or minimum value function to find the lowest point on the graph of the function.
3. The x-coordinate of this point will be the value of x that yields the smallest distance.
By following these steps, you can find the value of x that minimizes the distance from point P to the point (1, 2).