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? algebra 1b (flip over) →

Explanation:

Similarities:

• All three functions are smooth and continuous (defined for all real numbers in their standard forms: linear \( y = mx + b \), quadratic \( y = ax^2 + bx + c \), exponential \( y = a^x \) or \( y = ab^x \) with \( a>0, b>0, b

eq1 \)).

• They can all be used to model real - world relationships (e.g., linear for constant - rate change, quadratic for projectile motion, exponential for population growth/decay).

Differences:

Shape:

• Linear: Has a straight - line graph. The general form is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. The slope \( m \) determines the steepness and direction (positive \( m \) means increasing from left to right, negative \( m \) means decreasing).
• Quadratic: Has a parabola shape. The general form is \( y = ax^{2}+bx + c\) (\(a

eq0\)). If \( a>0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards. It has a vertex (a minimum or maximum point) and an axis of symmetry.

• Exponential: Has a curved graph that either grows rapidly or decays rapidly. For \( y = ab^{x}\), if \( b>1 \), it is an exponential growth function (the graph rises steeply as \( x \) increases), and if \( 0 < b < 1 \), it is an exponential decay function (the graph falls towards the x - axis as \( x \) increases).

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

• Linear: If \( m>0 \), as \( x\to\infty \), \( y\to\infty \) and as \( x\to-\infty \), \( y\to-\infty \); if \( m < 0 \), as \( x\to\infty \), \( y\to-\infty \) and as \( x\to-\infty \), \( y\to\infty \).
• Quadratic: If \( a>0 \), as \( x\to\pm\infty \), \( y\to\infty \); if \( a < 0 \), as \( x\to\pm\infty \), \( y\to-\infty \).
• Exponential: For \( y = ab^{x}\) with \( b>1 \): as \( x\to\infty \), \( y\to\infty \) and as \( x\to-\infty \), \( y\to0 \) (approaches the x - axis). For \( y = ab^{x}\) with \( 0 < b < 1 \): as \( x\to\infty \), \( y\to0 \) and as \( x\to-\infty \), \( y\to\infty \).

Rate of Change:

• Linear: Has a constant rate of change (the slope \( m \)). The change in \( y \) with respect to \( x \) is always \( m \), so \( \frac{\Delta y}{\Delta x}=m \) for any interval \( \Delta x \).
• Quadratic: Has a variable rate of change. The first derivative (in calculus terms) \( y^\prime=2ax + b \) is a linear function, which means the rate of change itself is changing. For example, for \( y=x^{2}\), the rate of change at \( x = 1 \) is \( 2(1)=2 \), and at \( x = 2 \) is \( 2(2) = 4 \).
• Exponential: Has a rate of change that is proportional to the function's value. The derivative of \( y = ab^{x}\) is \( y^\prime=ab^{x}\ln(b) \), so the rate of change \( \frac{dy}{dx}\) is \( \ln(b)\times y \), meaning that as the function's value \( y \) increases (for growth, \( b > 1 \)), the rate of change also increases.

Answer:

Similarities:

• All are smooth, continuous, and model real - world relationships.

Differences:

• Shape: Linear (straight line), Quadratic (parabola), Exponential (curved, growth/decay).
• End - behavior: Linear (depends on slope), Quadratic (depends on \( a \) sign), Exponential (depends on \( b \) value in \( y = ab^{x}\)).
• Rate of change: Linear (constant), Quadratic (variable, changing rate), Exponential (rate proportional to function value).