Axis Of Symmetry: Ball's Height Function Explained

Alright guys, let's break down how to find the axis of symmetry for the given height function. This is a classic quadratic equation problem, and understanding the axis of symmetry will give us some cool insights into the ball's trajectory. So, let's get started!

Understanding the Quadratic Function

Okay, first things first, let's make sure we're all on the same page. The given function is $h(t) = -16t^2 + 400$. This is a quadratic function in the form of $h(t) = at^2 + bt + c$, where:

• $a = -16$
• $b = 0$ (since there's no $t$ term)
• $c = 400$

This quadratic function represents the height $h(t)$ of the ball at time $t$. The coefficient $a = -16$ indicates that the parabola opens downwards, which makes sense because gravity is pulling the ball back down. The constant term $c = 400$ tells us that the initial height of the ball is 400 feet.

Axis of Symmetry Explained

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It's like a mirror – the parabola is symmetrical on either side of this line. For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the equation for the axis of symmetry is given by:

$x = -\frac{b}{2a}$

In our case, the variable is $t$ instead of $x$, and the function is $h(t) = -16t^2 + 400$. So, we'll use $t$ instead of $x$ in the formula. The axis of symmetry is crucial because it tells us at what time the ball reaches its maximum height. It's the point where the ball momentarily stops before changing direction and falling back down.

Calculating the Axis of Symmetry

Now, let's calculate the axis of symmetry for our function $h(t) = -16t^2 + 400$. We know that $a = -16$ and $b = 0$. Plugging these values into the formula, we get:

$t = -\frac{0}{2(-16)}$

$t = -\frac{0}{-32}$

$t = 0$

So, the equation for the axis of symmetry is $t = 0$. This means that the maximum height of the ball occurs at $t = 0$ seconds.

Interpreting the Result

The fact that the axis of symmetry is at $t = 0$ makes sense in the context of the problem. The ball is thrown upwards from a height of 400 feet. Since there's no initial upward velocity term (the $bt$ term is absent), the maximum height occurs at the initial time, $t = 0$. In other words, the ball is already at its highest point when it's released.

To further illustrate this, consider what the graph of $h(t) = -16t^2 + 400$ looks like. It's a parabola opening downwards, with its vertex (the highest point) on the y-axis (the $h(t)$-axis) at $t = 0$. This confirms that the axis of symmetry is indeed at $t = 0$.

Alternative Method: Completing the Square

Another way to find the axis of symmetry is by completing the square. This method transforms the quadratic function into vertex form, which directly reveals the vertex coordinates and, consequently, the axis of symmetry. Let's walk through it:

$h(t) = -16t^2 + 400$

Since there's no $t$ term, we can rewrite it as:

$h(t) = -16(t - 0)^2 + 400$

This is now in vertex form: $h(t) = a(t - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In our case, $h = 0$ and $k = 400$. The vertex is at $(0, 400)$, and the axis of symmetry is the vertical line that passes through the vertex, which is $t = 0$.

Benefits of Completing the Square

Completing the square is super useful because it not only helps you find the axis of symmetry but also gives you the vertex of the parabola directly. The vertex represents the maximum or minimum point of the function, which can be very helpful in optimization problems. Plus, it gives you a better understanding of the shape and position of the parabola.

Practical Implications

Understanding the axis of symmetry has practical applications in various fields, not just in textbook problems. For example:

• Physics: When analyzing projectile motion, the axis of symmetry helps determine the time at which the projectile reaches its maximum height.
• Engineering: In designing parabolic reflectors (like satellite dishes or solar collectors), the axis of symmetry is crucial for focusing energy or signals.
• Optimization: In business and economics, quadratic functions are often used to model costs or profits, and the axis of symmetry helps find the point of maximum profit or minimum cost.

Common Mistakes to Avoid

When finding the axis of symmetry, here are some common mistakes to watch out for:

• Incorrectly identifying $a$ and $b$: Make sure you correctly identify the coefficients $a$ and $b$ in the quadratic function. Remember, $a$ is the coefficient of the $t^2$ term, and $b$ is the coefficient of the $t$ term.
• Forgetting the negative sign: The formula for the axis of symmetry is $t = -\frac{b}{2a}$. Don't forget the negative sign in front of the fraction!
• Not simplifying correctly: Double-check your calculations to make sure you simplify the expression correctly.
• Confusing the axis of symmetry with the vertex: The axis of symmetry is a vertical line, while the vertex is a point. The axis of symmetry passes through the x-coordinate (or t-coordinate in our case) of the vertex.

Real-World Example

Let's say you're designing a water fountain that shoots water in a parabolic arc. The height of the water jet can be modeled by a quadratic function. By finding the axis of symmetry, you can determine at what horizontal distance the water jet reaches its maximum height. This information is crucial for designing the fountain so that the water lands where you want it to.

Conclusion

So, to wrap it up, the equation for the axis of symmetry of the given function $h(t) = -16t^2 + 400$ is $t = 0$. Remember, the axis of symmetry is a vertical line that passes through the vertex of the parabola, and it helps us understand the symmetry of the quadratic function. By correctly identifying the coefficients and using the formula, you can easily find the axis of symmetry and gain valuable insights into the problem. Keep practicing, and you'll become a pro at solving these types of problems! Understanding these concepts not only helps in math class but also gives you a foundation for tackling real-world problems in various fields. Keep up the great work, and remember to double-check your work to avoid common mistakes! Understanding the axis of symmetry is a fundamental concept in quadratic functions, offering insights into the behavior and symmetry of parabolas. Whether you're dealing with projectile motion, engineering designs, or optimization problems, mastering this concept will undoubtedly prove valuable. Keep honing your skills, and you'll be well-equipped to tackle a wide range of mathematical challenges. And that's a wrap!