Evaluating F(x) = 5x^2 - 45x + 101 A Step-by-Step Solution
This article provides a comprehensive guide on how to evaluate the quadratic function $f(x) = 5x^2 - 45x + 101$ for various inputs. We will explore how to find the function's value at specific points, including integers, fractions, and symbolic representations. Additionally, we will delve into determining the input values that result in a specific output, such as $f(x) = 1$. This step-by-step approach will equip you with the skills to confidently analyze and manipulate quadratic functions.
(a) Evaluating f(4)
To evaluate the function at $x = 4$, we substitute $4$ for $x$ in the function's expression. This process involves replacing every instance of $x$ with the value $4$. The key here is to follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculation. Carefully substituting and simplifying is crucial in obtaining the correct result. Quadratic functions, like this one, are fundamental in mathematics and often appear in various applications, making it essential to master their evaluation.
Therefore, to find $f(4)$, we perform the following calculation:
First, we calculate the square:
Next, we perform the multiplications:
Finally, we perform the additions and subtractions from left to right:
Thus, the value of the function $f(x)$ at $x = 4$ is $1$. This result indicates a specific point on the parabola represented by the quadratic function. Understanding how to evaluate a function at a specific point is crucial for graphing and analyzing the behavior of the function. The point (4,1) lies on the graph of the function and provides valuable information about the function's properties. This type of calculation forms the basis for more complex analysis, such as finding roots, vertex, and other key features of the quadratic function.
(b) Evaluating f(-1/2)
Now, let's evaluate the function at $x = - rac{1}{2}$. This involves substituting $- rac{1}{2}$ for $x$ in the function's expression. Working with fractions requires careful attention to detail, especially when dealing with squaring and multiplication. Accurate handling of fractions is essential to avoid errors. This evaluation will give us the function's value at a fractional input, demonstrating the function's behavior across a wider range of values. Evaluating at fractional inputs is important because it shows the function's continuity and helps in understanding its overall trend.
To find $f(- rac{1}{2})$, we perform the following calculation:
f(- rac{1}{2}) = 5(- rac{1}{2})^2 - 45(- rac{1}{2}) + 101
First, we calculate the square:
f(- rac{1}{2}) = 5( rac{1}{4}) + rac{45}{2} + 101
Next, we perform the multiplications:
f(- rac{1}{2}) = rac{5}{4} + rac{45}{2} + 101
To add these terms, we need a common denominator, which is $4$. So, we convert the fractions:
f(- rac{1}{2}) = rac{5}{4} + rac{90}{4} + rac{404}{4}
Now, we add the numerators:
f(- rac{1}{2}) = rac{5 + 90 + 404}{4}
f(- rac{1}{2}) = rac{499}{4}
Thus, the value of the function $f(x)$ at $x = - rac{1}{2}$ is $ rac{499}{4}$. This result gives us another point on the parabola, and its fractional value highlights the importance of being able to evaluate functions for non-integer inputs. The point (-1/2, 499/4) is another data point on the graph, and its location further defines the shape and position of the parabola. This calculation reinforces the concept that quadratic functions can take on a wide range of values depending on the input.
(c) Evaluating f(a)
Evaluating $f(a)$ involves substituting the variable $a$ for $x$ in the function's expression. This is a symbolic evaluation, meaning the result will be an expression in terms of $a$ rather than a numerical value. This process helps us understand the function's general behavior and how it transforms the input variable. Symbolic evaluation is a fundamental skill in algebra and is crucial for understanding function transformations and compositions.
To find $f(a)$, we simply substitute $a$ for $x$:
This simplifies to:
Therefore, the value of the function $f(x)$ at $x = a$ is $5a^2 - 45a + 101$. This expression represents the general form of the function for any input $a$. This symbolic representation is essential for further analysis, such as finding the derivative or integral of the function. The resulting expression is a quadratic in 'a', mirroring the original function's structure, but now with 'a' as the independent variable.
(d) Evaluating f(2/m)
Evaluating $f( rac{2}{m})$ involves substituting the expression $ rac{2}{m}$ for $x$ in the function's expression. This again results in a symbolic evaluation, where the output will be an expression in terms of $m$. This exercise highlights how functions can handle more complex inputs and how to manipulate algebraic expressions involving fractions and variables. Careful substitution and simplification are key to obtaining the correct result. This type of evaluation is common in various mathematical contexts, such as when dealing with transformations or composite functions.
To find $f( rac{2}{m})$, we perform the following substitution:
f( rac{2}{m}) = 5( rac{2}{m})^2 - 45( rac{2}{m}) + 101
First, we calculate the square:
f( rac{2}{m}) = 5( rac{4}{m^2}) - rac{90}{m} + 101
Next, we perform the multiplications:
f( rac{2}{m}) = rac{20}{m^2} - rac{90}{m} + 101
To combine these terms into a single fraction, we need a common denominator, which is $m^2$. So, we convert the fractions:
f( rac{2}{m}) = rac{20}{m^2} - rac{90m}{m^2} + rac{101m^2}{m^2}
Now, we combine the numerators:
f( rac{2}{m}) = rac{101m^2 - 90m + 20}{m^2}
Thus, the value of the function $f(x)$ at $x = rac{2}{m}$ is $ rac{101m^2 - 90m + 20}{m^2}$. This expression represents the function's output for a rational input, showcasing the versatility of function evaluation. The resulting expression is a rational function in 'm', derived from the quadratic nature of the original function and the rational input. This demonstrates how functions can transform inputs into more complex expressions.
(e) Finding Values of x such that f(x) = 1
To find the values of $x$ such that $f(x) = 1$, we need to solve the equation $5x^2 - 45x + 101 = 1$. This involves setting the function equal to $1$ and then solving the resulting quadratic equation. Solving quadratic equations is a fundamental skill in algebra, and several methods can be used, such as factoring, completing the square, or using the quadratic formula. Choosing the appropriate method depends on the specific equation.
First, we set the function equal to $1$:
Next, we subtract $1$ from both sides to set the equation equal to zero:
We can simplify the equation by dividing all terms by $5$:
Now, we can solve this quadratic equation. In this case, the equation is factorable. We look for two numbers that multiply to $20$ and add up to $-9$. These numbers are $-4$ and $-5$. So, we can factor the equation as:
Now, we set each factor equal to zero and solve for $x$:
x - 4 = 0$ or $x - 5 = 0
Solving these equations gives us:
x = 4$ or $x = 5
Therefore, the values of $x$ such that $f(x) = 1$ are $x = 4$ and $x = 5$. These values represent the x-coordinates where the parabola intersects the horizontal line $y = 1$. Finding these values is crucial for understanding the function's behavior and its relationship to specific output values. The solutions represent the points where the function's graph has a y-coordinate of 1, providing important insights into the function's range and intercepts.
In conclusion, this article has demonstrated how to evaluate the quadratic function $f(x) = 5x^2 - 45x + 101$ for various inputs, including integers, fractions, and symbolic representations. We have also shown how to find the input values that result in a specific output. These skills are essential for understanding and manipulating quadratic functions in various mathematical contexts.