To find the equation of a quadratic function given a graph, we need to use the vertex form of the quadratic equation, which is:
\[y = a(x-h)^2 + k\]
where (h, k) represents the vertex of the graph.
First Graph:
From the given points (-1, 1), (1, 1), and (4, 2), we notice that the graph is symmetrical and has a vertex at (1, 1). Therefore, we can determine that h=1 and k=1.
Substituting these values into the vertex form equation, we have:
\[y = a(x-1)^2 + 1\]
To find the value of 'a,' we need to use another point from the graph. Let's use (-2, 4):
\[4 = a(-2-1)^2 + 1\]
\[4 = a(-3)^2 + 1\]
\[4 = 9a + 1\]
\[9a = 4 - 1\]
\[9a = 3\]
\[a = \frac{1}{3}\]
So, the equation of the quadratic function for the first graph is:
\[y = \frac{1}{3}(x-1)^2 + 1\]
Second Graph:
From the given points (-2, 4) and (-4, 2), we can determine that the vertex is not provided in the given points. However, by analyzing the symmetry, we can see that the vertex must be halfway between the two given points, which would be at (-3, 3). Therefore, h=-3 and k=3.
Substituting these values into the vertex form equation, we have:
\[y = a(x+3)^2 + 3\]
Using either (-2, 4) or (-4, 2), let's use (-2, 4) as an example:
\[4 = a(-2+3)^2 + 3\]
\[4 = a(1)^2 + 3\]
\[4 = a + 3\]
\[a = 4 - 3\]
\[a = 1\]
So, the equation of the quadratic function for the second graph is:
Answers & Comments
Verified answer
To find the equation of a quadratic function given a graph, we need to use the vertex form of the quadratic equation, which is:
\[y = a(x-h)^2 + k\]
where (h, k) represents the vertex of the graph.
First Graph:
From the given points (-1, 1), (1, 1), and (4, 2), we notice that the graph is symmetrical and has a vertex at (1, 1). Therefore, we can determine that h=1 and k=1.
Substituting these values into the vertex form equation, we have:
\[y = a(x-1)^2 + 1\]
To find the value of 'a,' we need to use another point from the graph. Let's use (-2, 4):
\[4 = a(-2-1)^2 + 1\]
\[4 = a(-3)^2 + 1\]
\[4 = 9a + 1\]
\[9a = 4 - 1\]
\[9a = 3\]
\[a = \frac{1}{3}\]
So, the equation of the quadratic function for the first graph is:
\[y = \frac{1}{3}(x-1)^2 + 1\]
Second Graph:
From the given points (-2, 4) and (-4, 2), we can determine that the vertex is not provided in the given points. However, by analyzing the symmetry, we can see that the vertex must be halfway between the two given points, which would be at (-3, 3). Therefore, h=-3 and k=3.
Substituting these values into the vertex form equation, we have:
\[y = a(x+3)^2 + 3\]
Using either (-2, 4) or (-4, 2), let's use (-2, 4) as an example:
\[4 = a(-2+3)^2 + 3\]
\[4 = a(1)^2 + 3\]
\[4 = a + 3\]
\[a = 4 - 3\]
\[a = 1\]
So, the equation of the quadratic function for the second graph is:
\[y = (x+3)^2 + 3\]
Answer
...................................