Calculate Area Under y-4=-(x-3)²: Shifted Parabola Problem

Video Solution

Solution Steps

00:07 Let's find the positive domain of the function.

00:10 This means, where the graph is above the X-axis.

00:14 We'll use shortcut multiplication formulas and expand the brackets.

00:20 First, we'll arrange the equation as a function.

00:37 Next, collect like terms.

00:42 Notice the coefficient of X squared is negative.

00:46 A negative coefficient means the curve faces down.

00:51 Now, find where it crosses the X-axis.

00:58 At these points, Y equals zero. Substitute and solve.

01:05 Change from negative to positive.

01:12 Take the square root.

01:15 Remember, the square root gives two answers: positive and negative.

01:23 Solve each separately to find X.

01:38 These are where the graph meets the X-axis.

01:44 Now, let's draw the graph using these points and the curve's direction.

01:51 The graph is positive where it's above the X-axis.

01:57 And that's the solution to our problem! Great job!

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the critical points where the parabola is zero.
Step 2: Solve the inequality to determine where the parabola is above the x-axis.
Step 3: Use these results to specify the correct domain of x-values.

Step 1: Finding the critical points where the function is zero, solve:
$- (x-3)^2 + 4 = 0$ .

Re-organize to find:
$-(x-3)^2 = -4$ simplifies to $(x-3)^2 = 4$ .

Step 2: Take the square root of both sides:
$x-3 = \pm2$ .

This gives critical points $x = 3 + 2 = 5$ and $x = 3 - 2 = 1$ .

Step 3: Since the parabola opens downwards, the function is positive between these roots. Therefore, the interval where the function is positive is:

$1 < x < 5$ .

Therefore, the solution to the problem is $1 < x < 5$ .