How to Determine the Maximum Height of a Quadratic Function
Quick Answer
To find the maximum height of a quadratic function like -x² + 14x - 40, use the vertex formula x = -b/2a. Substituting values gives a maximum height of 9 feet.
When dealing with quadratic functions, one of the important concepts to understand is how to find the maximum or minimum value of the function, often represented graphically as the vertex of a parabola. In this case, we have the quadratic expression for the height of a rocket given as
$$-x^2 + 14x - 40$$.
This expression is a downward-opening parabola, meaning it has a maximum point. To find this maximum height, we can use the vertex formula, which is given by
$$x = -\frac{b}{2a}$$
where \(a\) and \(b\) are coefficients from the standard form of the quadratic equation, \(ax^2 + bx + c\).
In our case, we identify \(a = -1\), \(b = 14\), and \(c = -40\). Plugging these values into the vertex formula gives:
$$x = -\frac{14}{2(-1)} = 7.$$
This result indicates that the maximum height occurs when the horizontal distance traveled by the rocket is 7 feet. To find the maximum height of the rocket, we substitute \(x = 7\) back into the original equation:
$$-7^2 + 14(7) - 40$$
$$= -49 + 98 - 40$$
$$= 9.$$
Thus, the maximum height of the rocket is indeed 9 feet, confirming that the checked statement is correct.
Next, it’s important to know how to find the total horizontal distance the rocket travels from launch to landing. This occurs when the height of the rocket is equal to zero. We can find this by setting the quadratic equation to zero:
$$-x^2 + 14x - 40 = 0.$$
To solve for \(x\), we can factor or use the quadratic formula. Factoring gives:
$$(x - 10)(x - 4) = 0,$$
which results in the solutions \(x = 10\) and \(x = 4\). This means the rocket returns to the ground at these two points. The total horizontal distance it travels is the difference between these two values, which is 10 feet.
In conclusion, understanding how to analyze quadratic functions is crucial not only in math class but also in real-world scenarios, such as physics problems involving projectile motion. By mastering these concepts, you can confidently tackle various mathematical challenges involving parabolas.
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