Question

category 2 practice questions

1. easy

in the xy-plane, what is the y-coordinate of the y-intercept of the function f defined by f(x) = 3x⁴ - 4x³ + 55?

1. medium

the table above shows selected values of x and their corresponding f(x) values of the function f. what is the value of (f(2) - f(1))/f(4)?

x	f(x)
2	1
3	2
4	4

1. medium

in the xy-plane, the graph of the function f is shown above, where y = f(x). which of the following satisfies f(x) = 3?
a) f(-2) and f(2)
b) f(-2) and f(3)
c) f(-3)
d) f(3)

1. hard

in the xy-plane, the complete graph of the function h is shown above, and y = h(m). if h(m) = c has 4 solutions and c is a constant, what is one possible value of c?
a) -3
b) -2
c) 0
d) 1

1. hard

for the function g, three values of x and their corresponding values of g(x) are shown in the table above. if 2g(x) = g(3x) for all values of x, which of the following is the value of g(6)?
a) 3
b) 5
c) 6
d) 10

x	g(x)
4	-2
12	-4

Explanation:

Question 1:

Step1: Recall y - intercept definition

The y - intercept of a function \(y = f(x)\) is the value of \(y\) when \(x = 0\). So we need to find \(f(0)\) for the function \(f(x)=3x^{4}-4x^{3}+55\).

Step2: Substitute \(x = 0\) into the function

Substitute \(x = 0\) into \(f(x)\): \(f(0)=3(0)^{4}-4(0)^{3}+55\). Since any non - zero number to the power of \(0\) is \(1\), but \(0\) to any positive power is \(0\), so \(3(0)^{4}=0\), \(- 4(0)^{3}=0\). Then \(f(0)=0 - 0+55 = 55\).

Step1: Find \(f(2)\), \(f(1)\) and \(f(4)\) from the table

From the table, when \(x = 1\), \(f(1)=-3\); when \(x = 2\), \(f(2)=1\); when \(x = 4\), \(f(4)=4\).

Step2: Calculate \(f(2)-f(1)\)

\(f(2)-f(1)=1-(-3)=1 + 3=4\).

Step3: Calculate the fraction \(\frac{f(2)-f(1)}{f(4)}\)

Substitute the values we found: \(\frac{4}{4}=1\).

Step1: Analyze the graph of \(y = f(x)\)

We need to find the values of \(x\) for which \(f(x)=3\). Looking at the graph of \(y = f(x)\), we can see the \(y\) - value of \(3\) occurs at two points. We check the \(x\) - values corresponding to \(y = 3\). From the graph, when \(y = 3\), the \(x\) - values are \(x=-2\) and \(x = 2\) (since the graph has a peak at \(x=-2\) with \(y = 3\) and another peak at \(x = 2\) with \(y = 3\)). So \(f(-2)=3\) and \(f(2)=3\), which means \(f(-2)\) and \(f(2)\) satisfy \(f(x)=3\).

Answer:

The y - coordinate of the y - intercept is \(55\).

Question 2: