Problem 61 - Algebra 1

Question

EVALUATING FUNCTIONS Evaluate the function for these values of \(x:-2,-1,0,\) and \(1 .\) Organize your results in a table. $$ y=-x+12.1 $$

Step-by-Step Solution

Answer

The solutions are \((-2, 14.1)\), \((-1, 13.1)\), \((0, 12.1)\) and \((1, 11.1)\). These are represented in a table where each row matches a unique \(x\) value with its corresponding \(y\) value.

Step 1 - Substitute \(x = -2\)

Substitute \(x = -2\) into the equation \(y = -x + 12.1\) to get \(y = -(-2) + 12.1\), which simplifies to \(y = 2 + 12.1 = 14.1\). So the pair \((-2, 14.1)\) is a solution.

Step 2 - Substitute \(x = -1\)

Substitute \(x = -1\) into the equation \(y = -x + 12.1\) to get \(y = -(-1) + 12.1\), which simplifies to \(y = 1 + 12.1 = 13.1\). So the pair \((-1, 13.1)\) is a solution.

Step 3 - Substitute \(x = 0\)

Substitute \(x = 0\) into the equation \(y = -x + 12.1\) to get \(y = -0 + 12.1\), which simplifies to \(y = 12.1\). So the pair \((0, 12.1)\) is a solution.

Step 4 - Substitute \(x = 1\)

Substitute \(x = 1\) into the equation \(y = -x + 12.1\) to get \(y = -1 + 12.1\), which simplifies to \(y = 11.1\). So the pair \((1, 11.1)\) is a solution.

Step 5 - Summarize Results in a Table

Organize the solutions in a table. The table could be as follows:| \(x\) | \(y\)| --- | ---| -2 | 14.1| -1 | 13.1| 0 | 12.1| 1 | 11.1

Key Concepts

Substitution Method

The substitution method is a cornerstone of algebra, particularly useful when dealing with functions. It involves replacing a variable with a specific value, allowing us to evaluate the function at that point. Think of it like filling in the blanks: wherever you see the variable (usually 'x' or 'y'), you replace it with the given number. For example, if you have the function f(x) = 2x + 3 and you're asked for f(4), you'd substitute '4' into 'x' to get f(4) = 2(4) + 3 = 11.

When you're working with the substitution method, take careful note of minus signs and parentheses. Mistakes often occur when substituting negative numbers if you forget that the negative sign should change a plus to a minus, or vice versa, within the calculation.

Linear Functions

Linear functions describe a straight-line relationship between two variables, typically 'x' and 'y'. The general form of a linear equation is y = mx + b, where 'm' represents the slope or the steepness of the line, and 'b' stands for the y-intercept, which is where the line crosses the y-axis.

Understanding the graph of a linear function can help in visually grasping the concept of slope and intercepts. For instance, if 'm' is positive, the line slants upwards; if 'm' is negative, it slants downwards. The value of 'b' tells us the starting point of the line. Linear functions have the property of constant rate of change, which means that for a unit change in 'x', 'y' changes consistently by the value of 'm'.

Function Evaluation

Evaluating a function is all about determining the output value of the function given an input. It's like a machine; you put in a number (the input), the machine does something with that number according to a predefined rule (the function), and then out comes a result (the output).

Take the function y = -x + 12.1 as an example. When we evaluate this function for various values of 'x', we are essentially calculating what 'y' would be for each input. If you input '0' into this function, you get '12.1' out, because the rule says to multiply 'x' by '-1' and add '12.1'. The value we calculate is dependent solely on our input and the formula we're given—that's the essence of function evaluation.

Creating a Value Table

Creating a value table helps organize the input and output values of a function, aiding in understanding and visualization. To build a value table, list down all the input values (usually 'x') in a column, and then use the function to find and list down the corresponding output values ('y') in a neighboring column.

As seen in the textbook solution you're referring to, the value table shows a clear relationship between 'x' and 'y'. Organizing results in this way not only helps in keeping track of the numbers but can also serve as a basis for graphing the function, identifying patterns, and understanding the function's behavior. Remember, accuracy is key—double-check your calculations to ensure that the values in the table reflect the correct outputs from the function.