Question

If

q
(
x
)
is a linear function, where
q
(
−
1
)
=
3
, and
q
(
3
)
=
5
, determine the slope-intercept equation for
q
(
x
)
, then find q(2).

The equation of the line is:

q(2) =


If
t
(
x
)
is a linear function, where
t
(
−
4
)
=
3
, and
t
(
4
)
=
4
, determine the slope-intercept equation for
t
(
x
)
, then find t(0).

The equation of the line is:

t(0) =

Answer 1

`\large\boxed{Q1.\ q(x)=(1)/(2)x+(7)/(2),\ q(2)=(9)/(2)}\\\boxed{Q2.\ t(x)=(1)/(2)x+(7)/(2),\ t(0)=(7)/(2)}`

Explanation:

`\text{The slope-intercept form of an equation of a line:}\\\\y=mx+b\\\\m-slope\\b-y-intercept\\\\\text{The formula of a slope:}\\\\m=(y_2-y_1)/(x_2-x_1)\\\\\text{We have}\\\\q(-1)=3\to(-1,\ 3)\\q(3)=5\to(3,\ 5)`

`\text{Calculate the slope:}\\\\m=(5-3)/(3-(-1))=(2)/(4)=(2:2)/(4:2)=(1)/(2)\\\\\text{We have the equation:}\\\\y=(1)/(2)x+b\\\\\text{Put the coordinates of the point (-1, 3) to the equation:}\\\\3=(1)/(2)(-1)+b\\\\3=-(1)/(2)+b\qquad\text{add}\ (1)/(2)\ \text{to both sides}\\\\3(1)/(2)=b\to b=3(1)/(2)=(7)/(2)`

`q(x)=(1)/(2)x+(7)/(2)\\\\q(2)-\text{put x = 2 to the equation:}\\\\q(2)=(1)/(2)(2)+(7)/(2)=(2)/(2)+(7)/(2)=(9)/(2)`

`t(-4)=3\to(-4,\ 3)\\t(4)=4\to(4,\ 4)\\\\m=(4-3)/(4-(-4))=(1)/(8)\\\\y=(1)/(8)x+b\\\\\text{put the coordinates of the point (4,\ 4):}\\\\4=(1)/(8)(4)+b\\\\4=(1)/(2)+b\qquad\text{subtract}\ (1)/(2)\ \text{from both sides}\\\\3(1)/(2)=b\to b=3(1)/(2)=(7)/(2)\\\\t(x)=(1)/(2)x+(7)/(2)\\\\t(0)=(1)/(2)(0)+(7)/(2)=0+(7)/(2)=(7)/(2)`