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Chapter 8: Problem 55

Find the maximum or minimum value of \(y\) for each function. $$y=-3 x^{2}+14$$

Short Answer

Expert verified
The maximum value of \( y \) is 14 at \( x = 0 \).

Step by step solution

01

Identify the function's form

The given function is a quadratic function, which is generally written as \[ y = ax^2 + bx + c \] In this case, the function is \[ y = -3x^2 + 14 \].
02

Determine the coefficient 'a'

Here, the coefficient 'a' is -3. Since 'a' is less than 0, the parabola opens downwards and thus has a maximum value.
03

Find the vertex

The vertex of a parabola \( y = ax^2 + bx + c \) can be found using the formula \[ x = -\frac{b}{2a} \]. For the given equation, there is no 'b' term (or it is 0), thus \[ x = 0 \].
04

Calculate the y-coordinate

To find the maximum value, substitute \( x = 0 \) into the function \[ y = -3(0)^2 + 14 = 14 \].
05

State the maximum value

Therefore, the maximum value of \( y \) is 14 at \( x = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Maximum Value
In quadratic functions, the maximum or minimum value is found at the vertex of the parabola. Quadratic functions have the general form: where 'a', 'b', and 'c' are constants. The sign of the coefficient 'a' determines whether the parabola opens upwards or downwards. If 'a' is positive, the parabola opens upwards and has a minimum value at its vertex. Conversely, if 'a' is negative, the parabola opens downwards and has a maximum value at its vertex. In our given function, since 'a' is -3 (which is less than 0), the parabola opens downwards and therefore has a maximum value. To find this maximum value, we need to locate the vertex.
Parabola
A parabola is the graph of a quadratic function and has a distinctive 'U' shape. The direction in which the parabola opens depends on the coefficient 'a' in the quadratic function If 'a' is greater than 0, the parabola opens upwards. If 'a' is less than 0, the parabola opens downwards. The highest or lowest point on the parabola is called the vertex. This is where we find the function's maximum or minimum value. In our example, we were given which is an upside-down parabola because 'a' is -3. Hence, it has a maximum value at its vertex.
Vertex Formula
The vertex of a quadratic function can be found using the vertex formula where 'a' and 'b' are the coefficients from the quadratic equation. In our function the 'b' term is 0 (since there is no 'x' term), making our vertex located at To find the maximum value (the y-coordinate of the vertex), we substitute this x-value back into the function: Substituting we get: Therefore, the maximum value of for this function is 14. Using these principles in any quadratic function will help you find either the maximum or minimum value it can achieve.

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Most popular questions from this chapter

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$v^{2}=3 v+5$$

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$y^{2}-\frac{1}{2} y+\frac{1}{4}=0$$

Solve each inequality. State the solution set using interval notation when possible. \(16-x^{2}>0\)

Write a quadratic equation of each of the following types, then trade your equations with those of a classmate. Solve the equations and verify that they are of the required types. a) a single rational solution b) two rational solutions c) two irrational solutions d) two imaginary solutions

Solve each inequality. State the solution set using interval notation when possible. \(x^{2} \geq 0\)
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