PLZ HELP
Recall that a radical function can be expressed as f(x)= a*sqrt( x - h) + k. How do the values of A, H, and K, affect the domain and range of a radical function? How is this similar to other functions you have learned about (such as linear, quadratic, or exponential functions)? How is it different? Give examples and justify your ideas.
Thanks, but what does that have to do with the domain and range of a radical function?
since √x is only defined for x≥0, you need x≥h
and, since √x ≥ 0, that means that f(x) ≥ h if a > 0
plezzzz helps
To understand how the values of a, h, and k affect the domain and range of a radical function, let's break down each term:
1. "a" is the coefficient in front of the square root. It determines the vertical stretch or compression of the graph. If a > 1, the graph will be vertically stretched; if 0 < a < 1, it will be compressed. The domain and range remain the same regardless of the value of a.
2. "h" is the horizontal shift. It shifts the graph horizontally. The effect of h on the domain is opposite to its sign (positive or negative). If h is positive, the graph shifts to the right, reducing the domain; if h is negative, the graph shifts to the left, also reducing the domain. The range remains unaffected by the value of h, as it only affects the x-values.
3. "k" is the vertical shift. It shifts the graph vertically. The effect of k only affects the range. If k is positive, the graph shifts upward, increasing the range; if k is negative, the graph shifts downward, decreasing the range. The domain remains unchanged by the value of k.
To compare with other functions:
1. Linear functions (f(x) = mx + b) have a constant slope (m) and domain and range of all real numbers (-∞, ∞). The vertical shift (b) has a similar effect as the "k" value in the radical function, shifting the graph vertically.
2. Quadratic functions (f(x) = ax^2 + bx + c) have a vertical stretch or compression (a), a horizontal shift (h), and a vertical shift (k). Similar to the radical function, the horizontal shift (h) affects the domain, and the vertical shift (k) affects the range.
3. Exponential functions (f(x) = ab^x) have a horizontal shift (h) and a vertical shift (k). Similar to the radical function, the horizontal shift (h) affects the domain, and the vertical shift (k) affects the range.
However, the radical function is different in that it contains a square root, leading to the possibility of a limited domain. For example, the expression inside the square root must be greater than or equal to zero, otherwise, the function will generate imaginary numbers. This constraint restricts the domain of a radical function.
Let's consider an example: f(x) = 2√(x - 3) + 4.
- The coefficient "2" (a) indicates a vertical stretch.
- The horizontal shift (h) is "3", which shifts the graph three units to the right, reducing the domain.
- The vertical shift (k) is "4", shifting the graph four units upward, increasing the range.
Thus, the domain would be x ≥ 3, and the range would be y ≥ 4.