Problem 55 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \[ y = -3x^2 + 14 \].

Step 2: 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.

Step 3: 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 \[ x = 0 \].

Step 4: Calculate the y-coordinate

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

Step 5: State the maximum value

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

Key Concepts

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.