Question

3.1.2 discuss & practice
directions: think about the following question and respond in the space provided below. (10 pts)
what similarities and differences do you notice about the shape and behavior of the graphs for linear, quadratic, and exponential functions?

Explanation:

Similarities:

• All three functions are smooth and continuous (no breaks or holes) over their domains (usually all real numbers for basic forms).
• They can model relationships between two variables (e.g., input \( x \) and output \( y \)).

Differences:

Shape:

• Linear: Graph is a straight line (form: \( y = mx + b \), where \( m \) is slope, \( b \) is y - intercept).
• Quadratic: Graph is a parabola (U - shaped or inverted U - shaped; form: \( y = ax^2+bx + c \), \( a

eq0 \)). It has a vertex (minimum or maximum point).

• Exponential: Graph is a curve (form: \( y = ab^x \), \( a

eq0 \), \( b>0,b
eq1 \)). If \( b > 1 \), it increases rapidly (exponential growth); if \( 0 < b < 1 \), it decreases rapidly (exponential decay).

Behavior (as \( x\to\pm\infty \)):

• Linear: If \( m>0 \), \( y\to\infty \) as \( x\to\infty \) and \( y\to-\infty \) as \( x\to-\infty \); if \( m < 0 \), the opposite.
• Quadratic: If \( a>0 \), \( y\to\infty \) as \( x\to\pm\infty \); if \( a < 0 \), \( y\to-\infty \) as \( x\to\pm\infty \).
• Exponential: If \( b>1 \), \( y\to\infty \) as \( x\to\infty \) and \( y\to0 \) as \( x\to-\infty \); if \( 0 < b < 1 \), \( y\to0 \) as \( x\to\infty \) and \( y\to\infty \) as \( x\to-\infty \).

Rate of Change:

• Linear: Constant rate of change (slope \( m \) is constant).
• Quadratic: Rate of change (slope of tangent line) is linear (changes at a constant rate, since the derivative \( y'=2ax + b \) is linear).
• Exponential: Rate of change is proportional to the function's value (derivative of \( y = ab^x \) is \( y'=ab^x\ln(b) \), so it grows/shrinks by a percentage of its current value).

Answer:

Similarities:

• All are smooth, continuous, and model variable relationships.

Differences:

• Shape: Linear = straight line; Quadratic = parabola; Exponential = curve.
• Behavior at Infinity: Linear has linear growth/decay; Quadratic has symmetric end - behavior; Exponential has asymptotic or rapid growth/decay.
• Rate of Change: Linear = constant; Quadratic = linear rate of change; Exponential = rate proportional to itself.