Problem 24 - Algebra 1

Question

Make an input-output table for the function. Use 1, 1.5, 3, 4.5, and 6 as the domain. $$ y=1.5+x^{2} $$

Step-by-Step Solution

Answer

The input-output table for the function \(y=1.5+x^{2}\) is: x = [1, 1.5, 3, 4.5, 6] y = [2.5, 3.75, 10.5, 21.75, 37.5].

Step 1: Identify function

Identify the formula given for the function which is \(y=1.5+x^{2}\). This function tells us how to get an output (y) for every input (x).

Step 2: List the given inputs

List the given values of x which are 1, 1.5, 3, 4.5, and 6.

Step 3: Compute the output for each input

Substitute each x value into the function to solve for y. That means, calculate \(y=1.5+x^{2}\) for each x in the list.

Step 4: Calculate y when x=1

Substitute x=1 into the function to get \(y=1.5+1^{2}=1.5+1=2.5\). So when x=1, y=2.5.

Step 5: Calculate y when x=1.5

Substitute x=1.5 into the function to get \(y=1.5+(1.5)^{2}=1.5+2.25=3.75\). So when x=1.5, y=3.75.

Step 6: Calculate y when x=3

Substitute x=3 into the function to get \(y=1.5+3^{2}=1.5+9=10.5\). So when x=3, y=10.5.

Step 7: Calculate y when x=4.5

Substitute x=4.5 into the function to get \(y=1.5+(4.5)^{2}=1.5+20.25=21.75\). So when x=4.5, y=21.75.

Step 8: Calculate y when x=6

Substitute x=6 into the function to get \(y=1.5+6^{2}=1.5+36=37.5\). So when x=6, y=37.5.

Step 9: Write out the input-output table

Finally, to make an input-output table for the function, write the corresponding x (input) and y (output) values in tabular form: x = [1, 1.5, 3, 4.5, 6] y = [2.5, 3.75, 10.5, 21.75, 37.5].

Key Concepts

Quadratic Functions

Quadratic functions are quite special in the world of mathematics. At their core, they are polynomials that take a very specific form. A basic quadratic function is expressed as \[ y = ax^2 + bx + c \] where \( a, b, \) and \( c \) are constants and never equal zero for \( a \).

• The \( a \) term, often called the coefficient of \( x^2 \), determines how wide or narrow the parabola is.
• The \( b \) term can affect the direction of the parabola as it moves along the x-axis.
• The \( c \) term is the y-intercept of the parabola, showing where it crosses the y-axis.

In the exercise, the function given is \( y = 1.5 + x^2 \), where \( a = 1 \), \( b = 0 \), and \( c = 1.5 \). This means the parabola opens upwards, has a wide "U" shape, and crosses the y-axis at 1.5. Quadratic functions always create a U-shaped graph! The shape is called a "parabola". It's crucial to understand this shape, as it helps predict the behavior of the function.

Domain and Range

Domain and range are two foundational concepts that describe the inputs and outputs of a function.**Domain:**The domain of a function is the complete set of possible values of the independent variable, which is typically represented as \( x \). When you hear "domain," think of all the possible x values that you can put into the function.

• In our example, the domain is explicitly given as 1, 1.5, 3, 4.5, and 6.
• This means you can only use these values as inputs for \( x \).

**Range:**The range of a function is all the values that the function can output, represented by \( y \). It shows all the possible outcomes as you consider every value in the domain.

• For the quadratic function in this exercise, the outputs are the calculated y-values: 2.5, 3.75, 10.5, 21.75, and 37.5.

Understanding domain and range helps us know the limits of a function and its behavior over different values. This knowledge ensures we only work with suitable values when evaluating functions.

Function Evaluation

Evaluating a function simply means calculating its output for specific inputs. This is an essential part of working with any function because it's how you really see the function in action.To evaluate a function like \( y = 1.5 + x^2 \):

• Take a value from the domain – in this case, maybe \( x \) is 1.
• Plug this \( x \) value into the function to replace the variable: \( y = 1.5 + 1^2 \).
• Simplify the expression to find \( y \): here, \( y = 2.5 \).

Function evaluation allows you to construct the input-output table by finding what y-value corresponds to each x-value. It's a skill that helps with deeper understanding, especially when dealing with graphs of functions. It provides a concrete and straightforward way to visualize the relationship between variables. This process is not just applicable during school but is a valuable skill in all areas of mathematics, from solving equations to analyzing real-world data.