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Magazine Chapter 10: Problem 201
In the following exercises, find the maximum or minimum value. $$ y=-9 x^{2}+16 $$
Short Answer
Expert verified
The maximum value is 16.
Step by step solution
01
- Identify the function type
The given function is a quadratic equation of the form \( y = ax^2 + bx + c \). In this case, the equation is \( y = -9x^2 + 16 \).
02
- Determine the leading coefficient
Identify the leading coefficient, \( a \). Here, \( a = -9 \), which is less than zero. Since the coefficient is negative, the parabola opens downwards, indicating that the function has a maximum value.
03
- Find the vertex
For a quadratic function \( y = ax^2 + bx + c \), the x-coordinate of the vertex, where the maximum or minimum value occurs, is found using \( x = -\frac{b}{2a} \). In this equation, \( b = 0 \) and \( a = -9 \), giving \( x = -\frac{0}{2(-9)} = 0 \).
04
- Calculate the y-coordinate of the vertex
Substitute \( x = 0 \) back into the original equation to find the y-coordinate: \( y = -9(0)^2 + 16 = 16 \). This is the maximum value of the function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
maximum value
In a quadratic equation, the maximum or minimum value of the function is determined by the direction in which the parabola opens. For the equation given in the exercise,
$$ y = -9x^2 + 16 $$
the parabola opens downwards because the leading coefficient is negative.
When a parabola opens downwards, its highest point is called the maximum value. This maximum value is the y-coordinate of the vertex.
In our exercise, we found that the maximum value is 16. This tells us that no matter what value of x we choose, the function will never exceed 16.
Always remember, the shape of the parabola (upward or downward) gives us a clue about whether we are dealing with a maximum or minimum value.
$$ y = -9x^2 + 16 $$
the parabola opens downwards because the leading coefficient is negative.
When a parabola opens downwards, its highest point is called the maximum value. This maximum value is the y-coordinate of the vertex.
In our exercise, we found that the maximum value is 16. This tells us that no matter what value of x we choose, the function will never exceed 16.
Always remember, the shape of the parabola (upward or downward) gives us a clue about whether we are dealing with a maximum or minimum value.
vertex
The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. It is a crucial element in understanding and graphing the function.
To find the vertex of a quadratic equation in the form
$$ y = ax^2 + bx + c $$
we use the formulas:
x-coordinate:
$$ x = -\frac{b}{2a} $$
y-coordinate:
We substitute the x-coordinate back into the original equation.
In our example, with the equation
$$ y = -9x^2 + 16 $$
we identified b = 0 and a = -9. Substituting these values into the formula gives:
x = 0
To find the y-coordinate, we substitute x = 0 into the equation:
y = -9(0)^2 + 16 = 16
Thus, the vertex of the parabola is at (0, 16).
To find the vertex of a quadratic equation in the form
$$ y = ax^2 + bx + c $$
we use the formulas:
x-coordinate:
$$ x = -\frac{b}{2a} $$
y-coordinate:
We substitute the x-coordinate back into the original equation.
In our example, with the equation
$$ y = -9x^2 + 16 $$
we identified b = 0 and a = -9. Substituting these values into the formula gives:
x = 0
To find the y-coordinate, we substitute x = 0 into the equation:
y = -9(0)^2 + 16 = 16
Thus, the vertex of the parabola is at (0, 16).
leading coefficient
The leading coefficient is the coefficient of the term with the highest power (degree) in a polynomial. In a quadratic equation of the form
$$ y = ax^2 + bx + c $$
the leading coefficient is 'a'. It has a significant impact on the graph of the equation.
For the quadratic equation
$$ y = -9x^2 + 16 $$
the leading coefficient is -9.
Here鈥檚 what the leading coefficient tells us:
$$ y = ax^2 + bx + c $$
the leading coefficient is 'a'. It has a significant impact on the graph of the equation.
For the quadratic equation
$$ y = -9x^2 + 16 $$
the leading coefficient is -9.
Here鈥檚 what the leading coefficient tells us:
- If a > 0, the parabola opens upwards, and the function has a minimum value.
- If a < 0, the parabola opens downwards, and the function has a maximum value.
parabola
A parabola is a symmetrical, U-shaped curve represented by a quadratic equation of the form
$$ y = ax^2 + bx + c $$
The direction in which the parabola opens is determined by the sign of the leading coefficient 'a'.
In our quadratic function
$$ y = -9x^2 + 16 $$
the parabola opens downwards because the leading coefficient is -9.
Key features of a parabola include:
$$ y = ax^2 + bx + c $$
The direction in which the parabola opens is determined by the sign of the leading coefficient 'a'.
In our quadratic function
$$ y = -9x^2 + 16 $$
the parabola opens downwards because the leading coefficient is -9.
Key features of a parabola include:
- Vertex: The highest or lowest point on the graph, depending on whether the parabola opens down or up.
- Axis of symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images.
- Direction: Determined by the sign of 'a'. If 'a' is positive, it opens upwards; if 'a' is negative, it opens downwards.
