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A quadratic function is a type of polynomial that has an equation of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The defining feature of such functions is the presence of the squared term \( x^2 \).Quadratic functions produce a U-shaped curve known as a parabola when graphed on a coordinate plane. This means they have a highest or lowest point, referred to as the vertex. The parabola opens upwards if \( a > 0 \) or downwards if \( a < 0 \). This will determine whether the function has a minimum or maximum value.Understanding the properties of quadratic functions is fundamental, as they frequently occur in various areas of algebra and calculus. In the given function \( t(x) = -5x^2 + 3x \), the presence of \( -5x^2 \) indicates the function is quadratic and the negative coefficient tells us the parabola opens downwards.

The substitution method is a fundamental algebraic technique used to simplify and evaluate expressions. For evaluating functions, substitution involves replacing a variable (input value) with a given number within a function.Here's how it works step-by-step:

• Identify the function and the specific input values you are substituting, like \( x = -11 \) and \( x = 4 \) in our problem.
• Replace the variable \( x \) with the given input values, resulting in a simpler expression to calculate.

For example, substituting \( x = -11 \) in \( t(x) = -5x^2 + 3x \) involves replacing all instances of \( x \) with \(-11\), resulting in the expression \(-5(-11)^2 + 3(-11)\). This step transforms the algebraic expression into one that involves straightforward arithmetic operations, making the function easier to evaluate.

Output calculation involves performing arithmetic operations to determine the result of the function once a specific input value has been substituted.After substitution, you must carry out the arithmetic calculations:

• Begin by handling the exponentiation, such as calculating \((-11)^2 = 121\).
• Next, perform multiplication: calculate \(-5 \times 121 = -605\) and \(3 \times (-11) = -33\).
• Finally, add or subtract the results, here: \(-605 - 33 = -638\).

This process is repeated for each given input, like \( x = 4 \) in the second part. Depending on the complexity of the numbers, rounding answers may be necessary to reflect more precision, like to three decimal places if specified by the problem. Learning output calculation in quadratic functions aids in handling various mathematical problems with accuracy.