Write a possible equation for the following graph using transformations. Point is (-2,0) and (0,2)
One possible equation for the graph can be expressed as:
y = a(x - h)² + k
where (h, k) represents the vertex of the parabola. We can determine the values of a, h, and k using the given points (-2,0) and (0,2).
Let's start with the x-coordinate of the vertex, which is h. We can find it as the average of the x-coordinates of the given points:
h = (x₁ + x₂) / 2
h = (-2 + 0) / 2
h = -1
Similarly, we find the y-coordinate of the vertex, k, as the average of the y-coordinates of the given points:
k = (y₁ + y₂) / 2
k = (0 + 2) / 2
k = 1
Now we can substitute the values of h and k into the equation to get:
y = a(x - (-1))² + 1
y = a(x + 1)² + 1
Next, we can use one of the points, (-2,0), to determine the value of a. Substituting the coordinates into the equation yields:
0 = a((-2) + 1)² + 1
0 = a(1)² + 1
0 = a + 1
Solving for a, we subtract 1 from both sides:
a = -1
Substituting the value of a back into the equation, we get the final equation:
y = -1(x + 1)² + 1
Therefore, one possible equation for the given graph is y = -1(x + 1)² + 1.
To find a possible equation for the graph using transformations, we need to determine the transformations applied to the basic function. The basic function for a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept.
Given the points (-2,0) and (0,2), we can find the slope using the slope formula:
m = (y2 - y1) / (x2 - x1)
= (2 - 0) / (0 - (-2))
= 2 / 2
= 1
Since the slope is 1, we have the equation y = x + b. To find the value of b, we can substitute one of the given points.
Using the point (-2,0):
0 = -2 + b
b = 2
Therefore, the equation for the given graph is y = x + 2.
To write an equation for the given graph using transformations, we need to start with a base equation and then apply appropriate transformations to it. In this case, we can start with a basic linear equation, y = mx + b, where m represents the slope and b represents the y-intercept.
Given that the graph goes through the points (-2, 0) and (0, 2), we can find the slope, m, using the formula:
m = (y2 - y1)/(x2 - x1)
Substituting the values, we get:
m = (2 - 0)/(0 - (-2))
m = 2/2
m = 1
Now, using the point-slope form of a linear equation, y - y1 = m(x - x1), we can substitute one of the given points and the calculated slope to get an equation:
Using the point (-2, 0):
y - 0 = 1(x - (-2))
y = x + 2
Thus, the equation for the given graph using transformations is y = x + 2.