Question

Given \(g(x)=3 x^{2}-7\) find (a) \(g(3 t)\) (c) \(g(6 t-4)\) (d) \(g(4 x+9)\)

Short answer

Question: Evaluate the function \(g(x) = 3x^2 - 7\) at the following points: a) \(g(3t)\) c) \(g(6t-4)\) d) \(g(4x+9)\). Answer: a) The expression for \(g(3t)\) is \(27t^2 - 7\). c) The expression for \(g(6t-4)\) is \(108t^2 - 144t + 41\). d) The expression for \(g(4x+9)\) is \(48x^2 + 216x + 236\).

Step-by-step solution

(a) Find g(3t)

To find \(g(3t)\), replace \(x\) with \(3t\) in the given function: \(g(3t) = 3(3t)^2 - 7\) Now, simplify this expression: \(g(3t) = 3(9t^2) - 7\) \(g(3t) = 27t^2 - 7\) The expression for \(g(3t)\) is \(27t^2 - 7\).

(c) Find g(6t-4)

To find \(g(6t-4)\), replace \(x\) with \(6t-4\) in the function \(g(x)\): \(g(6t-4) = 3(6t-4)^2 - 7\) Now, simplify this expression: \(g(6t-4) = 3(36t^2 - 48t + 16) - 7\) \(g(6t-4) = 108t^2 - 144t + 48 - 7\) \(g(6t-4) = 108t^2 - 144t + 41\) The expression for \(g(6t-4)\) is \(108t^2 - 144t + 41\).

(d) Find g(4x+9)

To find \(g(4x+9)\), replace \(x\) with \(4x+9\) in the given function: \(g(4x+9) = 3(4x+9)^2 - 7\) Now, simplify this expression: \(g(4x+9) = 3(16x^2 + 72x + 81) - 7\) \(g(4x+9) = 48x^2 + 216x + 243 - 7\) \(g(4x+9) = 48x^2 + 216x + 236\) The expression for \(g(4x+9)\) is \(48x^2 + 216x + 236\).

Key concepts

Understanding Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
For example, the function given in the exercise is a quadratic polynomial: \(g(x) = 3x^2 - 7\).
This has the highest exponent of 2, making it a quadratic expression. Polynomial functions can take various forms, notably:

• Linear polynomials (e.g., \(ax + b\)): where the degree is 1.
• Quadratic polynomials (e.g., \(ax^2 + bx + c\)): the degree is 2, as seen in this exercise.
• Cubic polynomials (e.g., \(ax^3 + bx^2 + cx + d\)): these have a degree of 3.

Polynomial functions are fundamental in mathematics because they can model quadratic, linear, and other types of relationships, which is crucial for different fields of study.
This exercise explores simplifying polynomial expressions by plugging in different expressions for \(x\) to observe how the function changes.

The Art of Function Substitution

Function substitution is a method where we replace the variable with another expression or value.
This is seen in the given exercise, where \(x\) is replaced with expressions such as \(3t\) or \(6t-4\) in \(g(x) = 3x^2 - 7\).

It involves two main steps:

• Substitute the variable: Replace \(x\) in the original function with a new expression, seen as "inserting" new values into the function's formula.
• Simplify the expression: Simplify the results to uncover the new polynomial expression.
  This means applying basic algebra to reduce any multiplications, raising powers, and arithmetic operations.

This process helps visualize how changing your input affects the outcome, thus aiding in understanding the behavior of the function under different conditions.
It's also a helpful tool across math-related disciplines to solve complex problems and understand functions deeper.

Exploring Quadratic Expressions

Quadratic expressions have the general form \(ax^2 + bx + c\), where the "2" in the exponent signifies the highest power of the variable.
These expressions represent parabolic graphs when plotted, which can open upwards or downwards based on the sign of the leading coefficient (\(a\)).

In the exercise, the expression \(g(x) = 3x^2 - 7\) is used in a substitution framework.

• Vertex of the parabola: For a standard quadratic expression \(ax^2 + bx + c\), the vertex gives us the turning point and can be found using the formula \(x = -\frac{b}{2a}\).
• Simplification technique: Applying substitution and simplifying like \(g(6t-4)\), demonstrates changes in quadratic relations.
• Outcomes and transformations: Each substitution results in a new specific quadratic expression, like \(27t^2 - 7\) or \(108t^2 - 144t + 41\), showing transformations in the quadratic curve.
  This makes understanding these expressions essential for graphing, solving equations, and even physics applications.

Understanding how to manipulate quadratic expressions through substitution strengthens problem-solving skills in algebra and is foundational to mastering more complex mathematical concepts.