Question

the table represents the linear function f(x), and the equation represents the linear function g(x). g(x)=3x + 1 compare the y - intercepts and slopes of the linear functions f(x) and g(x) and choose the answer that best describes them. a the slope of f(x) is less than the slope of g(x). the y - intercept of f(x) is equal to the y - intercept of g(x). b the slope of f(x) is less than the slope of g(x). the y - intercept of f(x) is greater than the y - intercept of g(x). c the slope of f(x) is greater than the slope of g(x). the y - intercept of f(x) is equal to the y - intercept of g(x). d the slope of f(x) is greater than the slope of g(x). the y - intercept of f(x) is greater than the y - intercept of g(x).

Explanation:

Step1: Find slope and y - intercept of g(x)

The equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For \(g(x)=3x + 1\), the slope \(m_g=3\) and the y - intercept \(b_g = 1\).

Step2: Find slope of f(x) using slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,1)\) and \((x_2,y_2)=(2,9)\). Then \(m_f=\frac{9 - 1}{2-0}=\frac{8}{2}=4\).

Step3: Find y - intercept of f(x)

The y - intercept of \(f(x)\) is the value of \(y\) when \(x = 0\). From the table, when \(x = 0\), \(f(0)=1\), so \(b_f=1\).

Step4: Compare slopes and y - intercepts

The slope of \(f(x)\) is \(m_f = 4\) and the slope of \(g(x)\) is \(m_g=3\), so the slope of \(f(x)\) is greater than the slope of \(g(x)\). The y - intercept of \(f(x)\) is \(b_f = 1\) and the y - intercept of \(g(x)\) is \(b_g=1\), so the y - intercept of \(f(x)\) is equal to the y - intercept of \(g(x)\).

Answer:

The slope of \(f(x)\) is greater than the slope of \(g(x)\). The y - intercept of \(f(x)\) is equal to the y - intercept of \(g(x)\) (corresponding to option b).