Understanding the Maximum Height of a Parabola: A Step-by-Step Solution

When analyzing quadratic functions, one essential concept is identifying the vertex, which represents the maximum or minimum point of the parabola. In cases where the parabola opens downward (i.e., the coefficient of $x^2$ is negative), the vertex corresponds to the highest point — the maximum height.

This article walks through a clear, step-by-step solution to find the maximum value of the quadratic function $ y = -x^2 + 6x - 8 $.

Understanding the Context

Step 1: Recognize the Standard Form

The given quadratic equation is in standard form:

$$
y = ax^2 + bx + c
$$

Solution: The maximum height occurs at the vertex. For $ y = -x^2 + 6x - 8 $, $ a = -1 $, $ b = 6 $, so $ x = -\frac{6}{2(-1)} = 3 $. Substitute $ x = 3 $: $ y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1 $. The maximum height is $ 1 $ unit. \boxed{1}

Key Insights

Here,

  • $ a = -1 $
  • $ b = 6 $
  • $ c = -8 $

Since $ a < 0 $, the parabola opens downward, confirming a maximum value exists at the vertex.

Step 2: Calculate the x-Coordinate of the Vertex

The x-coordinate of the vertex is found using the formula:

Continue Reading

Matériel : \( 3x + 2y \leq 90 \)
Nous résolvons ces inégalités pour trouver des solutions faisables.
Résoudre \( 2x + 4y \leq 100 \) :

🔗 Related Articles You Might Like:

📰 Unhinged Secrets: Charlie Manson’s Hidden Messages That Changed America Forever 📰 The Dark Truth About Charlie Manson That Officials Won’t Let You Know! 📰 🌲 The Secret Behind Charlie Brown’s Tree: You Won’t Believe How It Changed the Entire Comic!

Final Thoughts

$$
x = - rac{b}{2a}
$$

Substitute $ a = -1 $ and $ b = 6 $:

$$
x = - rac{6}{2(-1)} = - rac{6}{-2} = 3
$$

So, the vertex occurs at $ x = 3 $.

Step 3: Substitute to Find the Maximum y-Value

Now plug $ x = 3 $ back into the original equation to find $ y $:

$$
y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1
$$

Thus, the maximum height is $ y = 1 $ unit.