Determine the Symmetry Point of f(x) = 5x - x^2

Video Solution

Solution Steps

00:00 Find the point of symmetry in the function

00:03 The point of symmetry is the point where if you fold the parabola in half

00:06 The halves will be equal to each other

00:12 Let's look at the function coefficients

00:22 We'll use the formula to calculate the vertex point

00:28 We'll substitute appropriate values according to the given data and solve for X at the point

00:36 This is the X value at the point of symmetry

00:39 Now we'll substitute this X value in the function to find the Y value at the point

00:56 This is the Y value at the point of symmetry

00:59 And this is the solution to the question

Step-by-Step Solution

To find the symmetry point of the quadratic function $f(x) = 5x - x^2$ , we will determine the vertex of the parabola.

Step 1: Express the quadratic function in standard form:
The given function is already in standard form: $f(x) = -x^2 + 5x$ , where $a = -1$ and $b = 5$ .
Step 2: Apply the vertex formula to find the x-coordinate of the vertex:
For the quadratic function $ax^2 + bx + c$ , the x-coordinate of the vertex is found using $x = -\frac{b}{2a}$ .
Step 3: Calculate the x-coordinate:
$\begin{aligned} x &= -\frac{5}{2 \times (-1)} \\ &= -\frac{5}{-2} \\ &= \frac{5}{2} \end{aligned}$
Step 4: Substitute $x = \frac{5}{2}$ back into the function to find the y-coordinate:
$\begin{aligned} f\left(\frac{5}{2}\right) &= -\left(\frac{5}{2}\right)^2 + 5\left(\frac{5}{2}\right) \\ &= -\frac{25}{4} + \frac{25}{2} \\ &= -\frac{25}{4} + \frac{50}{4} \\ &= \frac{25}{4} \end{aligned}$
Step 5: Determine the symmetry point:
The symmetry point, and thus the vertex of the function, is $\left(\frac{5}{2}, \frac{25}{4}\right)$ , or $(2\frac{1}{2}, 6\frac{1}{4})$ .

Therefore, the symmetry point of the function is $(2\frac{1}{2}, 6\frac{1}{4})$ .