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A quadratic function is a type of algebraic expression that includes an unknown variable squared, such as an equation of the form \( ax^2 + bx + c \). This means that there are three components: a quadratic term (\( ax^2 \)), a linear term (\( bx \)), and a constant term (\( c \)). In our exercise, the function we are working with is \( f(r) = -2r^2 + 1 \). Here, the quadratic term is \(-2r^2\) and the constant term is \(+1\). Notice there is no linear term in this function, which makes it a simpler form of quadratic function.
A key feature of quadratic functions is their graph, which takes the shape of a parabola. This can open upwards or downwards depending on the coefficients involved. For our specific function, the coefficient of the squared term \(-2\) is negative, which means the parabola opens downwards.

An input-output table is a convenient way to organize and visualize the results of substituting different values into a function. In our context, the input is the value we assign to the variable \( r \) within the quadratic function , and the output is the result of performing the calculations in the function based on this input.
Creating such a table helps us understand the relationship between inputs and outputs, providing a clear picture of how the function behaves as the input changes. For example, when we input different values like \(-1.7\), \(0.9\), and \(5.4\) into the function \( f(r) = -2r^2 + 1 \), we obtain unique outputs that tell us about the nature of the parabola and the impact each input value has.

The substitution method is a fundamental algebraic technique used to find an output by replacing a variable within an expression or function with a specific value. To apply this method to evaluate a function like \( f(r) = -2r^2 + 1 \), we follow these steps:

• Pick a specific value for the input variable (e.g., \( r = -1.7 \)).
• Replace the variable \( r \) in the function with this value, creating a new expression.
• Perform the arithmetic operations: This includes squaring the value, multiplying by the constant outside the squared term, and adding the constant term.
• Calculate the final result to discover the corresponding output.

Using the substitution method step-by-step ensures that each part of the process is understood, leading to accurate results and deeper insights into how different parts of the function contribute to the final output.

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. In our exercise, we are analyzing \( f(r) = -2r^2 + 1 \), which is a classic example of an algebraic expression involving both constants and variables.
This expression features a coefficient \(-2\) multiplied by the square of the variable \( r \), showing the impact of squaring \( r \) and then scaling it by \(-2\). Finally, \(+1\) is added to adjust the output.
Algebraic expressions like this one are the building blocks for functions and equations in algebra. They give us the framework to model real-world phenomena and solve problems. Understanding how to manipulate and evaluate these expressions is essential for progressing in mathematics.