What is the maximum vertical distance between the line y=x+2 and the parabola y=x^(2) for -1

Question

What is the maximum vertical distance between the line  y=x+2 and the parabola  y=x^(2) for  -1 <= x <= 2 ?

Bytelearn · Accepted Answer

Calculate Difference Function: To find the maximum vertical distance, we need to calculate the difference between the y-values of the line and the parabola for any x in the given interval. Difference function: d(x) = |(x + 2) - x^2|Simplify Difference Function: Simplify the difference function to make it easier to work with. d(x) = |x + 2 - x^2| = |-x^2 + x + 2|Find Vertex of Parabola: To find the maximum value of d(x), we need to find the vertex of the parabola represented by d(x), since the vertex will give us the maximum or minimum value of the function. The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex.Rewrite in Vertex Form: First, we need to rewrite d(x) in vertex form. To do this, we complete the square. d(x) = -(x^2 - x) + 2 Add and subtract (1/2)^2 inside the parenthesis to complete the square. d(x) = -(x^2 - x + (1/2)^2) + 2 + (1/2)^2Identify Parabola Vertex: Simplify the equation by combining like terms and factoring. d(x) = -(x - 1/2)^2 + 2 + 1/4 d(x) = -(x - 1/2)^2 + 9/4Find Maximum Value: Now that we have the vertex form, we can identify the vertex of the parabola. The vertex is at (h, k) = (1/2, 9/4). Since the coefficient of the squared term is negative, the parabola opens downwards, and the vertex represents the maximum point.Check Vertex Interval: The maximum value of d(x) is the y-coordinate of the vertex, which is 9/4. This is the maximum vertical distance between the line and the parabola within the given interval.Final Maximum Vertical Distance: However, we need to check if the vertex falls within the interval -1 <= x <= 2. Since h = 1/2, it does fall within the interval.Therefore, the maximum vertical distance between the line y=x+2 and the parabola y=x^2 for -1 <= x <= 2 is 9/4.

What is the maximum vertical distance between the line $y=x+2$ and the parabola $y=x^{2}$ for $-1 \leq x \leq 2$ ?

Full solution

More problems from Reflections of functions

Related topics

What is the maximum vertical distance between the line y=x+2 y=x+2 y=x+2 and the parabola y=x2 y=x^{2} y=x2 for −1≤x≤2 -1 \leq x \leq 2 −1≤x≤2 ?

What is the maximum vertical distance between the line $y=x+2$ and the parabola $y=x^{2}$ for $-1 \leq x \leq 2$ ?