Answer:
[tex](a)\ y = 2x +8[/tex]
[tex](b)\ y = 0[/tex]
[tex](c)\ y = -5[/tex]
Step-by-step explanation:
Given
[tex]y = x^2[/tex]
First, graph [tex]y = x^2[/tex] (See attachment)
We are not limited to any particular linear equation; as long as we meet the required condition
Solving (a): Equation of a line that intersects [tex]y = x^2[/tex] at two points
Select any two points on opposite sides of the curve
The selected points are:
[tex](x_1,y_1) = (-2,4)[/tex]
[tex](x_2,y_2) = (4,16)[/tex]
Calculate slope (m)
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]m = \frac{16 - 4}{4 - -2}[/tex]
[tex]m = \frac{12}{6}[/tex]
[tex]m = 2[/tex]
The equation is then calculated using:
[tex]y = m(x - x_1) + y_1[/tex]
This gives:
[tex]y = 2(x - -2) + 4[/tex]
[tex]y = 2(x +2) + 4[/tex]
Open bracket
[tex]y = 2x +4 + 4[/tex]
[tex]y = 2x +8[/tex]
Solving (b): Equation of a line that intersects [tex]y = x^2[/tex] at one point
A good illustration of this is x-axis
Because it intersects with the line at just one point, which is origin (0,0).
The equation of line at this point is:
[tex]y = 0[/tex]
Solving (c): Equation of a line that does not intersect [tex]y = x^2[/tex]
Here, we make use of any horizontal line below the x-axis
Select any two horizontal points below the x-axis
[tex](x_1,y_1) = (-5,-5)[/tex]
[tex](x_2,y_2) = (5,-5)[/tex]
Calculate slope
[tex]m = \frac{-5--5}{5--5}[/tex]
[tex]m = \frac{0}{10}[/tex]
[tex]m = 0[/tex]
The equation is then calculated using:
[tex]y = m(x - x_1) + y_1[/tex]
[tex]y = 0(x - -5) + (-5)[/tex]
[tex]y = 0-5[/tex]
[tex]y = -5[/tex]
