Question

Below are two different functions, f(x) and g(x). What can be determined about their y-intercepts?


f(x) = 4x − 1

chart:
x g(x)
1 3
4 6
7 9

Answer Choices:
a. The function f(x) has a higher y-intercept.
b. The function g(x) has a higher y-intercept.
c. They both have the same y-intercept.
d. The relationship between y-intercepts cannot be determined

Answer 1

For this case, the first thing to do is find the equation of the line for g (x).
 We have then:
 The equation of the line in point-slope form is given by:
 $y-yo = m (x-xo)$
 Where,
 (xo, yo): point that belongs to the line
 m: slope of the line
 The slope is given by:
 $m = \frac{y2-y1}{x2-x1}$
 Substituting values:
 $m = \frac{6-3}{4-1}$
 Rewriting:
 $m = \frac{3}{3}$
 $m=1$
 now we choose an ordered pair:
 $(xo, yo) = (1, 3)$
 Substituting values we have:
 $y-3=(x-1)$
 We now rewrite the equation in its standard form:
 $y=x-1+3$
 $y=x+2$
 Thus, the function g (x) is:
 $g(x)=x+2$
 On the other hand, the function f (x) is:
 $f(x)=4x-1$
 We note that the intersection with the y axis for g (x) is 2.
 The intersection with the y axis for f (x) is -1.
 Answer:
 b. The function g(x) has a higher y-intercept.

Answer 2

It can be determined about their y-intercepts that b. The function g(x) has a higher y-intercept.

Further explanation

Straight-line equations are mathematical equations that are described in the plane of cartesian coordinates

General formula

$\large{\boxed{\bold{y-y1=m(x-x1)}}}$

or

y = mx + c

Where

m = straight-line gradient which is the slope of the line

x1, y1 = the Cartesian coordinate that is crossed by the line

c = constant

The formula for a gradient (m) between 2 points in a line

m = Δy / Δx

$m=\frac{y_2-y_1}{x_2-x_1}$

To find the equation of the line which has the point of intersection with the highest line of y then we enter the value of x in the equation = 0, so we know which value of y is greater

In the problem there is one known line equation (f (x)) and one line equation known for the value of x and its function (g (x))

For equation f (x) the value of the intersection with the y axis is: (x = 0)

f (x) = 4x - 1 (f (x) = y)

y = 4x - 1

y = 4.0 - 1

y = -1

For equation g (x), we first look for the equation by finding the gradient first

we specify the points: (4, 6) and (7, 9)

then the price gradient (m)

$m=\frac{9-6}{7-4}$

m = 3/3

m = 1

We enter in the general equation with point (4,6)

y-6 = 1 (x-4)

y-6 = x-4

y = x + 2

y = g (x) = x + 2

We enter the value x = 0 to get the value of the intersection with the y axis:

y = x + 2

y = 2

These two results show that: b. The function g (x) has a higher y-intercept.

Learn more

F (x) = x2 + 1 g (x) = 5 - x

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the inverse of the function f (x) = 2x-10

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domain of the function

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Keywords: function, y-intercept.