Problem 63 - Algebra 1: Concepts and Skills

Question

Evaluate the function when x 0, 1, 2, 3, and 4. \(f(x)=-x^{2}\)

Step-by-Step Solution

Answer

The function values are: f(0)=0, f(1)=-1, f(2)=-4, f(3)=-9, f(4)=-16

Step 1: Substitute x=0 into the function

Plugging x=0 into the function gives \(f(0)=-0^{2}\), which simplifies to 0.

Step 2: Substitute x=1 into the function

Plugging x=1 into the function gives \(f(1)=-1^{2}\), which simplifies to -1.

Step 3: Substitute x=2 into the function

Substituting x=2 into the function gives \(f(2)=-2^{2}\), which simplifies to -4.

Step 4: Substitute x=3 into the function

Plug in x=3 into the function to get \(f(3)=-3^{2}\), which simplifies to -9.

Step 5: Substitute x=4 into the function

Lastly, putting x=4 into the function gives \(f(4)=-4^{2}\), which simplifies to -16.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra where you find the value of a function given a specific input. Essentially, if you're given a function, say, f(x), and you want to find the value when x is a particular number, you're performing function evaluation. This is akin to following a recipe where x represents the ingredients you have, and f(x) is the resulting dish.

For example, if a function is described as f(x) = -x^2 and you want to evaluate this function for x = 0, you would replace every instance of x in the function with 0, resulting in f(0) = -(0)^2 = 0. This process is repeated for any value you wish to substitute into the function. By systematically evaluating f(x) at different values of x, you can gain insight into the behavior of the function across a range of inputs.

Substitution in Functions

Substitution in functions is a technique where you replace the variable in a function with a particular value or another expression. In the context of the exercise, when evaluating f(x) = -x^2 at various values of x, you're applying substitution to find out what f(x) equals when x is 0, 1, 2, 3, or 4.

It's important to pay close attention to the order of operations when substituting. First, replace the variable with the given number, and then carry out the calculation remembering to follow the math rules like exponents before multiplication or addition. For instance, when x is substituted with 2, the sequence of operations is squaring 2 first, to get 4, and then applying the negative sign, resulting in f(2) = -4. This systematic replacement is key in understanding how the function behaves for different inputs and is a foundational skill for solving more complex equations.

Quadratic Functions

Quadratic functions, such as f(x) = -x^2 in our exercise, are a type of polynomial function with a degree of 2, which means the highest exponent of the variable x is 2. The standard form of a quadratic function is ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0.

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient a. In the example of f(x) = -x^2, since a is negative, the parabola opens downwards. By evaluating the function at multiple points, as done in the steps provided, you can plot the corresponding values on a graph to visualize the parabola. Quadratic functions have extensive applications in physics, engineering, and economics, as they model various phenomena including projectile motion and profit maximization.