Understanding Functions and Their Transformations

The degree of a polynomial function is determined by the highest power of the variable in the equation. In the function f(x) = 3x^2 + 2x + 1, the term with the highest exponent is 3x^2, which has a degree of 2. Since the degree is 2, this classifies the function as quadratic. Quadratic functions are characterized by their parabolic shape and include any polynomial of degree 2.

To find f(4) when f(x) = 2x + 3, substitute 4 for x in the function. This gives f(4) = 2(4) + 3. First, calculate 2 times 4, which equals 8. Then, add 3 to this result: 8 + 3 equals 11. Therefore, f(4) evaluates to 11.

The vertex of the function g(x) = (x - 2)^2 + 3 is derived from its standard form, where the vertex is at (h, k) with h = 2 and k = 3. This indicates that the graph is a parabola opening upwards, shifted 2 units to the right and 3 units up from the origin. The transformations reflect the horizontal and vertical shifts applied to the basic quadratic function y = x^2, confirming that the vertex is indeed (2, 3) with the specified translations.

A vertical shift in a function occurs when a constant is added to or subtracted from the function's output. In this case, adding 4 to the function f(x) shifts the entire graph of the function upwards by 4 units. This means every point on the graph moves up by 4, resulting in a vertical shift. Conversely, subtracting 4 would shift it down, while altering the input (like f(x + 4) or f(x - 4)) affects the horizontal position, not the vertical.

To find the x-intercepts of the function f(x) = x^2 - 4, we set the equation equal to zero: x^2 - 4 = 0. This can be factored into (x - 2)(x + 2) = 0. Solving for x gives us two solutions: x = 2 and x = -2. Therefore, the function has two x-intercepts, which are the points where the graph crosses the x-axis.

To derive the equation of the parabola, we start with the standard form \( f(x) = (x - h)^2 + k \). The transformations indicate a reflection over the x-axis, which introduces a negative sign, and shifts to the right by 3 units (h = 3) and down by 2 units (k = -2). Therefore, the equation becomes \( f(x) = -(x - 3)^2 - 2 \), accurately reflecting the transformations applied to the vertex of the parabola.

To determine the type of relationship, we first calculate the first differences of the y-values: 5 - 2 = 3 and 10 - 5 = 5. The first differences are 3 and 5. Next, we find the second differences: 5 - 3 = 2. Since the second differences are constant (equal to 2), this indicates a quadratic relationship, as quadratic functions have constant second differences. Therefore, the relationship between x and y in the given values is quadratic.

The function h(x) = 1/(x - 1) is defined for all real numbers except where the denominator equals zero. Setting the denominator x - 1 to zero gives x = 1, which would make the function undefined. Therefore, the domain includes all real numbers except for 1, as this value results in division by zero, leading to an undefined expression.

The vertical line test is a visual method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the relation fails the test, indicating that it is not a function. This is because, by definition, a function can have only one output (y-value) for each input (x-value). Therefore, the vertical line test is a quick way to assess the functional nature of a graph.

To find the value of y when x = 3 in the equation y = 2x + 1, substitute 3 for x, resulting in y = 2(3) + 1 = 6 + 1 = 7. The domain of the equation is all real numbers because x can take any value. Similarly, the range is also all real numbers since y can take any value based on the linear relationship defined by the equation. Thus, for x = 3, y is 7, and both the domain and range encompass all real numbers.