Write an equation for the quadratic graphed below
x
-intercepts: (-1,0) and (3,0).
y
-intercept: (0,1)
1
2
3
4
5
x-intercepts: (-1,0) and (3,0)
y = a(x+1)(x-3)
That means y(0) = a(1)(-3) = -3a = 1
y = -1/3 (x+1)(x-3)
Sure, let's find the equation for this quadratic graph. We know that the x-intercepts are (-1,0) and (3,0), and the y-intercept is (0,1).
To find the equation, we can start by using the standard form of a quadratic equation: y = ax^2 + bx + c.
Since the x-intercepts are given, we can say that when x = -1 and x = 3, y is equal to 0. Plugging these values into the equation, we get:
0 = a(-1)^2 + b(-1) + c (equation 1)
0 = a(3)^2 + b(3) + c (equation 2)
Now, let's use the y-intercept, which is (0,1):
1 = a(0)^2 + b(0) + c (equation 3)
This equation simplifies to:
1 = c
Now, we have a system of three equations:
0 = a - b + c (equation 1)
0 = 9a + 3b + c (equation 2)
1 = c (equation 3)
Since we already know that c = 1, we can substitute this value into equations 1 and 2:
0 = a - b + 1 (equation 1)
0 = 9a + 3b + 1 (equation 2)
Now, let's solve this system of equations to find the values of a and b.
From equation 1, we can rewrite it as:
b = a + 1
Substituting this into equation 2, we get:
0 = 9a + 3(a + 1) + 1
0 = 9a + 3a + 3 + 1
0 = 12a + 4
Simplifying further:
12a = -4
a = -4/12
a = -1/3
Now that we know the value of a, we can substitute it back into equation 1 to find the value of b:
b = (-1/3) + 1
b = 2/3
Finally, we have the values of a and b: a = -1/3, and b = 2/3.
Therefore, the equation for the quadratic graph with x-intercepts (-1, 0) and (3, 0), and y-intercept (0, 1) is:
y = (-1/3)x^2 + (2/3)x + 1
To find an equation for the quadratic graphed below with x-intercepts (-1,0) and (3,0), as well as a y-intercept (0,1), we can use the vertex form of a quadratic equation:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
Since the x-intercepts are given, we can determine the vertex by finding the midpoint between the two x-intercepts. The midpoint formula is:
h = (x1 + x2)/2
k = y-intercept
Using the given x-intercepts (-1,0) and (3,0), we can calculate:
h = (-1 + 3)/2 = 2/2 = 1
The y-intercept is given as (0,1), so k = 1.
Substituting these values into the vertex form of the equation:
y = a(x - 1)^2 + 1
To find the value of a, we can use the y-intercept (0,1). Plugging in x = 0 and y = 1 into the equation, we get:
1 = a(0 - 1)^2 + 1
1 = a + 1
Solving for a, we find:
a = 0
Therefore, the equation of the quadratic is:
y = (0)(x - 1)^2 + 1
Simplifying this equation, we get:
y = 1
So the equation for the quadratic graphed below is y = 1.
To write the equation for a quadratic graph, we can start by using the vertex form of a quadratic equation, which is given by:
y = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex.
In this case, the x-intercepts are given as (-1,0) and (3,0), which means the graph touches the x-axis at these points, hence the y-coordinate of the vertex will be 0.
The y-intercept is given as (0,1), meaning the graph touches the y-axis at this point. Therefore, the value of k is 1.
Now, let's find the x-coordinate of the vertex. The x-coordinate of the vertex lies between the x-intercepts, which are -1 and 3. Therefore, we can find the average between these two values to get the x-coordinate of the vertex:
x-coordinate of vertex = (x-intercept1 + x-intercept2)/2
= (-1 + 3)/2
= 2/2
= 1
Now we have the vertex coordinates: (h, k) = (1, 0)
Substituting these values into the vertex form equation, we get:
y = a(x - 1)^2 + 0
Next, we can use the y-intercept to find the value of 'a'. The y-intercept represents the point (0,1) on the graph. Substituting these values into the equation:
1 = a(0 - 1)^2 + 0
1 = a(1)^2
1 = a
So, the value of 'a' is 1.
Finally, substituting 'a' into the equation, we get:
y = 1(x - 1)^2 + 0
Simplifying the equation, we have:
y = (x - 1)^2
Thus, the equation for the quadratic graph is y = (x - 1)^2.