Question

For each function, i) state the parameters \(a, b, h,\) and \(k\) ii) describe the transformation that corresponds to each parameter iii) sketch the graph of the function iv) identify the domain, range, equation of the horizontal asymptote, and any intercepts a) \(y=2(3)^{x}+4\) b) \(m(r)=-(2)^{r-3}+2\) c) \(y=\frac{1}{3}(4)^{x+1}+1\) d) \(n(s)=-\frac{1}{2}\left(\frac{1}{3}\right)^{\frac{1}{4} s}-3\)

Short answer

For a): (a=2, b=3, h=0, k=4), Domain: (-∞, ∞), Range: (4, ∞)For b): (a=-1, b=2, h=3, k=2), Domain: (-∞, ∞), Range: (-∞, 2)For c): (a=1/3, b=4, h=-1, k=1), Domain: (-∞, ∞), Range: (1, ∞)For d): (a=-1/2, b=(1/3)^{1/4}, h=0, k=-3), Domain: (-∞, ∞), Range: (-∞, -3)

Step-by-step solution

Analyze Function a

Given function: a) y = 2(3)^x + 4**i. Identifying Parameters:**- Exponential function of the form y = a * b^(x - h) + k- Here, a = 2, b = 3, h = 0, k = 4**ii. Transformation Description:**- 'a' represents vertical stretch by factor of 2.- 'b' indicates a base function growth of 3.- 'h' is the horizontal shift, 0 means no shift.- 'k' is vertical shift upwards by 4 units.**iii. Graph Sketch:**- This is an increasing function starting from (0,6) and approaching a horizontal asymptote as x tends to negative infinity at y = 4.**iv. Domain, Range, etc:** - Domain: (-∞, ∞)- Range: (4, ∞)- Horizontal asymptote: y = 4- Intercept: y-intercept at (0, 6)

Analyze Function b

Given function: b) m(r) = -(2)^(r-3) + 2**i. Identifying Parameters:**- Exponential function of the form y = a * b^(x - h) + k- Here, a = -1, b = 2, h = 3, k = 2**ii. Transformation Description:**- 'a' represents vertical reflection and stretch by factor of 1.- 'b' indicates a base function growth of 2.- 'h' is horizontal shift to the right by 3 units.- 'k' is vertical shift upwards by 2 units.**iii. Graph Sketch:**- This is a decreasing function starting from (3, 1) and approaching a horizontal asymptote as r tends to positive infinity at y = 2.**iv. Domain, Range, etc:** - Domain: (-∞, ∞)- Range: (-∞, 2)- Horizontal asymptote: y = 2- Intercept: y-intercept at (0, 2 - 8 = -6)

Analyze Function c

Given function: c) y = 1/3 * (4)^(x+1) + 1**i. Identifying Parameters:**- Exponential function of the form y = a * b^(x - h) + k- Here, a = 1/3, b = 4, h = -1, k = 1**ii. Transformation Description:**- 'a' represents vertical compression by factor of 1/3.- 'b' indicates a base function growth of 4.- 'h' is horizontal shift to the left by 1 units.- 'k' is vertical shift upwards by 1 units.**iii. Graph Sketch:**- This is an increasing function starting from (-1, 1.333) and approaching a horizontal asymptote as x tends to negative infinity at y = 1.**iv. Domain, Range, etc:** - Domain: (-∞, ∞)- Range: (1, ∞)- Horizontal asymptote: y = 1- Intercept: y-intercept at (0, 1 + 4/3 = 5/3)

Analyze Function d

Given function: d) n(s) = - 1/2 * (1/3)^(1/4s) - 3**i. Identifying Parameters:**- Exponential function of the form y = a * b^(x - h) + k- Here, a = -1/2, b = (1/3)^1/4, h = 0, k = -3**ii. Transformation Description:**- 'a' represents vertical reflection and compression by factor of 1/2.- 'b' indicates a base function with a decay factor of (1/3)^1/4.- 'h' is a horizontal shift, which in this case is none (0).- 'k' is vertical shift downwards by 3 units.**iii. Graph Sketch:**- This is a decreasing function starting from (0, -3.5) and approaching a horizontal asymptote as x tends to positive infinity at y = -3.**iv. Domain, Range, etc:** - Domain: (-∞, ∞)- Range: (-∞, -3)- Horizontal asymptote: y = -3- Intercept: y-intercept at (0, -3 - 1/2 = -7/2)

Key concepts

Transformation of Functions

Transformations in functions help us understand how a base graph can be shifted, stretched, or reflected in various ways. For exponential functions of the form \(y = a \times b^{(x - h)} + k\), each parameter \(a, b, h,\) and \(k\) plays a unique role in shaping the graph.
Here's what each parameter does:

• \(a\) affects the vertical stretch or compression and reflection. If \(a > 1\), the graph stretches vertically. If \(0 < a < 1\), the graph compresses. A negative \(a\) reflects the graph over the x-axis.
• \(b\) is the base of the exponential function and it determines the rate of growth or decay. If \(b > 1\), the function grows exponentially. If \(0 < b < 1\), the function decays exponentially.
• \(h\) influences the horizontal shift. Positive \(h\) shifts the graph to the right, while negative \(h\) shifts it to the left.
• \(k\) translates the graph vertically. Positive \(k\) shifts the graph upwards whereas negative \(k\) shifts it downwards.

Understanding these transformations can help you easily sketch graphs and analyze their behavior.

Horizontal Asymptotes

Horizontal asymptotes play a key role in understanding the behavior of exponential functions as the input values grow very large or very small. They represent the value that the graph of the function approaches but never actually reaches.
For an exponential function \(y = a \times b^{(x - h)} + k\), the horizontal asymptote is determined by the parameter \(k\).

• If \(k > 0\), the graph will approach the line \(y = k\) as \(x\) approaches either positive or negative infinity, depending on the function's growth or decay.
• In other cases where the function includes reflections or shifts, the value of \(k\) still dictates the horizontal asymptote. For instance, in the function \(n(s) = -\frac{1}{2} \times (\frac{1}{3})^{\frac{1}{4}s} - 3\), the horizontal asymptote is \(y = -3\).

This concept helps in predicting the long-term behavior of the function without needing to compute complex points. It simplifies the task of sketching and understanding the entire function.

Domain and Range

Knowing the domain and range of a function is crucial for understanding where the function exists and what values it can take.
The **domain** of an exponential function \(y = a \times b^{(x - h)} + k\) is typically all real numbers because exponential functions are defined for every real input value.
**Domain Examples:**

• For the function \(y = 2 \times 3^x + 4\), the domain is \((-\infty, \infty)\).
• For the function \(m(r) = -(2)^{r-3} + 2\), the domain is \((-\infty, \infty)\).

The **range** of these functions is affected by the vertical shifts and reflections.

• For instance, \(y = 2 \times 3^x + 4\) has a range of \((4, \infty)\) because \(3^x\) is always positive and when multiplied by 2 and shifted up by 4, it will always be greater than 4.
• In contrast, \(m(r) = -(2)^{r-3} + 2\) has a range of \(( - \infty, 2)\) because negative values push the function's output downwards, and it will always be below 2.

Fully understanding the domain and range allows one to precisely know where the function lives and ensures all possible values are covered.

Intercepts

Intercepts are points where the function crosses the axes. For exponential functions, identifying intercepts can be straightforward if we understand parameter transformations.
**Y-Intercept:** The point where the graph crosses the y-axis (i.e., when \(x = 0\)).

• For \(y = 2 \times 3^x + 4\), setting \(x = 0\) gives \(y = 2 \times 3^0 + 4 = 6\). So, the y-intercept is at \((0, 6)\).
• For \(m(r) = -(2)^{r-3} + 2\), set \(r = 0\), \(y = -(2)^{-3} + 2 = 2 - \frac{1}{8} = \frac{15}{8}\). So, the y-intercept is at \((0, -6)\).

**X-Intercept:** Solving for \(x\) when \(y = 0\) for these functions can be trickier, and often results in logarithmic solutions. Sometimes exponential functions won't have x-intercepts due to their asymptotic nature.
Finding intercepts provides key points for graphing and further aids in understanding the function's structure.