Problem 43 - Algebra and Trigonometry

Question

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$ f(x)=5 x^{2}-5 x $$

Step-by-Step Solution

Answer

a. The function has a minimum value since the leading coefficient of the quadratic term is positive. \n b. The minimum value is -0.25 and it occurs at x=0.5. \n c. The domain of the function is all real numbers, \(-\infin < x < \infin\), and the range is \(-0.25 \leq f(x) < \infin\).

Step 1: Identify if the function has a minimum or maximum value

For a quadratic function given in the form \( f(x)=ax^2+bx+c \), the sign of the a(the coefficient of x squared) determines whether the function attains a maximum or a minimum. If a > 0, the function has a minimum because the parabola opens upwards. If \(a<0\), the function has a maximum because the parabola opens downwards. Here, \(a=5\), which is greater than 0, therefore our function has a minimum value.

Step 2: Find the minimum value and where it occurs

To find the minimum or maximum value of the function, we utilize the formula for the vertex of a parabola, which is \(-\frac{b}{2a}\). Here, \(b=-5\) and \(a=5\) so our x-coordinate for the vertex is \(\frac{-(-5)}{2*5} = 0.5\). Substituting x=0.5 into our function we get \(f(0.5)= 5*(0.5)^2-5*0.5=-0.25\). Hence, the function obtains its minimum value of -0.25 at x=0.5.

Step 3: Identifying the function's domain and range

The domain of any quadratic function is all real numbers since the function is defined for any real value of x. Therefore, the domain is \(-\infin < x < \infin\). As for the range, as we established earlier, the function has a minimum value and opens upwards meaning that its range will be greater than or equal to its minimum point. Hence, the range is \(-0.25 \leq f(x) < \infin\)

Key Concepts

Parabola

The graph of a quadratic function forms a U-shaped curve known as a parabola. It can open either upwards or downwards, depending on the orientation dictated by its equation. If the parabola opens upward, it implies the function has a minimum point, while if it opens downward, the function has a maximum point.
Understanding the direction of the parabola is crucial when analyzing the function's behavior.

• An upward-opening parabola suggests that the values of the function decrease to a point and then increase. This structure is determined when the coefficient of the squared term is positive.
• A downward-opening parabola suggests that the values of the function increase to a point and then decrease. This occurs when the coefficient of the squared term is negative.

In essence, the parabola's orientation helps in deducing whether the quadratic function reaches a maximum or minimum value.

Vertex

The vertex of a parabola is its turning point, the highest or lowest spot on the graph depending on its orientation.The vertex provides critical information about the function's minimum or maximum values:

• The x-coordinate of the vertex can be calculated using the formula \(-\frac{b}{2a}\) from the quadratic equation \(f(x)=ax^2+bx+c\).
• Once we have the x-coordinate, we substitute it back into the function to find the corresponding y-coordinate, which represents either the minimum or maximum value.

For example, in the quadratic function \(f(x)=5x^2-5x\), the vertex is calculated to be at \((0.5,-0.25)\). This implies the lowest point (since the parabola opens upwards) is when \(x=0.5\), and the function's value at this point is \(-0.25\).
This functionality makes the vertex not only a crucial point on the graph but also a beneficial tool in understanding the function's optimal values.

Domain and Range

In mathematical terms, the domain is the set of all possible inputs for a function, while the range is the set of all possible outputs.With quadratic functions:
The domain is generally "all real numbers" because you can substitute any value for \(x\) in the quadratic formula without limitation.Thus, for the function \(f(x)=5x^2-5x\), the domain is:

• The entire set of real numbers: \(-\infty < x < \infty\).

However, determining the range requires more analysis since it's based on the parabola's direction and its vertex.

• In our example, because the parabola opens upwards and its vertex is the lowest point at \(-0.25\), the range is all values \(f(x)\) can achieve from \(-0.25\) upwards, or \(-0.25 \leq f(x) < \infty\).

This analysis aids in recognizing the span of values the function covers over its graph.

Minimum Value

The minimum value of a quadratic function is the lowest point on the graph of the parabola, which occurs at the vertex when the parabola opens upwards.

• To find the minimum value, first determine the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\).
• Next, plug this x-coordinate back into the function to get the corresponding y-coordinate, which is the minimum value.

Specifically, for the function \(f(x)=5x^2-5x\):

• The x-coordinate of the vertex is \(0.5\), and the function value at this x-coordinate is \(-0.25\), which is the minimum value of the function.

This concise process shows how the properties of quadratic equations make it easy to pinpoint critical points like the minimum value, aiding in practical applications and deeper mathematical understanding.

Coefficient of x squared

The coefficient of the \(x^2\) term in a quadratic function plays a vital role in determining the parabola's shape and orientation.

• A positive coefficient means the parabola opens upwards, and a minimum value is present.
• A negative coefficient indicates the parabola opens downwards, leading to a maximum value.

In our function \(f(x)=5x^2-5x\):

• The coefficient of \(x^2\) is 5, which is positive, so the parabola opens upwards with a minimum value.

This coefficient affects how "steep" or "wide" the parabola is. Larger values mean more steepness, while smaller values promote wider shapes.Understanding this coefficient's influence helps in quickly determining the basic nature of the quadratic function's graph, without requiring extensive calculation or graphing.