Problem 33 - Algebra and Trigonometry

Question

\(29-38=\) Find the maximum or minimum value of the function. $$ f(s)=s^{2}-1.2 s+16 $$

Step-by-Step Solution

Answer

The minimum value of the function is 15.64.

Step 1: Recognize the Form of the Function

The given function is a quadratic function of the form \( f(s) = as^2 + bs + c \), where \( a = 1 \), \( b = -1.2 \), and \( c = 16 \). Quadratic functions have graphs that are parabolas, which can open upwards or downwards.

Step 2: Determine the Direction of the Parabola

Check the sign of \( a \) to determine the direction in which the parabola opens. Since \( a = 1 > 0 \), the parabola opens upwards, indicating that the function has a minimum value.

Step 3: Calculate the Vertex

The vertex of the parabola gives the minimum or maximum value of the function. For a quadratic function \( as^2 + bs + c \), the \( s \)-coordinate of the vertex is given by the formula \( s = -\frac{b}{2a} \). Substituting \( b = -1.2 \) and \( a = 1 \) gives \( s = -\frac{-1.2}{2 \times 1} = 0.6 \).

Step 4: Find the Minimum Value

Substitute the \( s \)-coordinate of the vertex back into the function to find the minimum value. Substitute \( s = 0.6 \) into \( f(s) = s^2 - 1.2s + 16 \) to get \( f(0.6) = (0.6)^2 - 1.2(0.6) + 16 = 0.36 - 0.72 + 16 = 15.64 \).

Key Concepts

Parabolas

Parabolas are the U-shaped graphs that come from quadratic functions. A quadratic function typically looks like this: \( f(x) = ax^2 + bx + c \). It forms a parabola on a graph. This shape can open upwards or downwards depending on the value of \( a \) in the function.
If \( a \) is positive, the parabola opens upwards like a smile 😊. If \( a \) is negative, it opens downwards like a frown 🙁. This direction is important because it shows us if the quadratic function has a minimum value (when it opens upwards) or a maximum value (when it opens downwards).
Understanding the direction of a parabola helps us quickly know what kind of extreme value (max or min) the function might have.

Vertex of a Parabola

The vertex of a parabola is a special point that is either the lowest or the highest point on the graph. It’s where the parabola changes direction. This point is crucial because it can tell us the minimum or maximum value of the quadratic function.
The formula to find the vertex is very handy. For a quadratic equation like \( ax^2 + bx + c \), the vertex’s \( x \)-coordinate is found using \( x = -\frac{b}{2a} \).

• Calculate \( s = -\frac{b}{2a} \) using the given function's coefficients.
• This gives the location along the x-axis where the vertex is.
• Finding the y-coordinate involves substituting this value back into the original function.

Once you have both coordinates, you have the exact location of the vertex.
In our example, for the function \( f(s) = s^2 - 1.2s + 16 \), the vertex happens at \( s = 0.6 \), which you can substitute back into the equation to find the corresponding output of the function.

Minimum Value of a Function

Finding the minimum value of a function is all about understanding the vertex when the parabola opens upwards. Remember, this makes the vertex the lowest point on the graph.
To find the minimum value for a quadratic function that opens upwards, you substitute the x-value of the vertex into the function. This is the output or y-value of the vertex. It's as simple as plugging in the x-coordinate of the vertex back into the function.

• Calculate the vertex using the formula \( s = -\frac{b}{2a} \).
• Substitute \( s \) back into the original quadratic function.
• Solve to find the y-coordinate, which gives the minimum value.

For the function \( f(s) = s^2 - 1.2s + 16 \), substituting the vertex \( s = 0.6 \) gives you the minimum value: \( f(0.6) = 15.64 \).
This means 15.64 is the smallest value that this function can take.