Question

successive differences:
first successive differences are equal: ____ function
second successive differences equal: ____ function
ratios of successive y-values are all equal and r ≠ 1: ____ function.
graph: curve labeled exponential (circled), with quadratic, linear below
table: x (2, 3, 4, 7) | f(x) (2, 4, 6, 12)
function: ( f(x) = 3^x - 4 ), with linear, exponential, quadratic labels (classification table)

Explanation:

Step1: Recall function type properties

• Linear functions have constant first differences (slope is constant).
• Quadratic functions have constant second differences (the difference of differences is constant).
• Exponential functions have constant ratios of successive \( y \)-values (each \( y \)-value is a constant multiple of the previous one, \( r

eq1 \)).

Step2: Match properties to function types

• For "First Successive Differences are Equal": This matches the property of a Linear function (since linear functions have constant slope, so first differences are constant).
• For "Second Successive Differences are Equal": This matches the property of a Quadratic function (the second differences of a quadratic function \( ax^{2}+bx + c \) are constant \( 2a \)).
• For "Ratios of Successive \( y \)-values are all equal and \( r

eq1 \)": This matches the property of an Exponential function (exponential functions are of the form \( f(x)=a\cdot r^{x}+k \), so the ratio of \( f(x + 1)/f(x)=r \) when \( k = 0 \), and similar for non - zero \( k \) in the limit of successive values).

Answer:

First Successive Differences are Equal: \(\boldsymbol{\text{Linear}}\) function
Second Successive Differences are Equal: \(\boldsymbol{\text{Quadratic}}\) function
Ratios of Successive \( y \)-values are all equal and \( r
eq1 \): \(\boldsymbol{\text{Exponential}}\) function