the graph of a quadratic function has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine an expression for the function.
first of all the function has to be
y = a(x+1)(x-3) but we don't know what a is
the axis of symmetry and thus the vertix is midway between the x-intercepts.
So from your information the vertex is (1,4)
then
4 = a(1+1)(1-3)
a = -1
so y = -(x+1)(x-3) or y = -x^2 + 2x + 3
Sure, I'll gladly help you with that! To find the quadratic function that satisfies those conditions, we should start by finding the equation in vertex form, which is:
f(x) = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola. We know that the x-intercepts are -1 and 3, so our roots are x = -1 and x = 3. Therefore, the factors of the quadratic function are (x + 1) and (x - 3). Now let's determine the vertex.
The x-coordinate of the vertex is given by the average of the x-intercepts:
h = (-1 + 3)/2 = 2/2 = 1
To find the y-coordinate (k) of the vertex, we use the fact that the range consists of all numbers less than or equal to 4. This tells us that the maximum value of the function occurs at the vertex, which is at y = 4:
k = 4
So, the equation for our quadratic function becomes:
f(x) = a(x - 1)^2 + 4
Now we just need to find the value of 'a.' To do that, we can substitute one of the x-intercepts in the equation. Let's use x = -1:
0 = a(-1 - 1)^2 + 4
0 = 4a + 4
-4 = 4a
a = -1
Therefore, the expression for the quadratic function is:
f(x) = -(x - 1)^2 + 4
To determine an expression for the quadratic function, we need to find the vertex of the graph. The vertex of a quadratic function in the form f(x) = a(x - h)^2 + k is given by (h, k).
Given that the x-intercepts are -1 and 3, we know that the quadratic function crosses the x-axis at these points, which means the roots are -1 and 3. Therefore, we can factor the quadratic equation in the following form:
f(x) = a(x - (-1))(x - 3) = a(x + 1)(x - 3)
Now, we need to find the value of 'a' to determine the range of the function. We are given that the range consists of all numbers less than or equal to 4. The vertex of the quadratic function determines the highest or lowest point.
Since the highest point should have a y-coordinate of 4, we have:
f(x) = a(x + 1)(x - 3)
To find the value of 'a,' we substitute the coordinates of either x-intercept into the equation:
f(-1) = a(-1 + 1)(-1 - 3) = 4
0 = a(0)(-4)
0 = 0
Since the equation becomes 0 = 0 when substituted with either x-intercept, it indicates that the highest or lowest point is at 0. Thus, a = 4.
Therefore, the expression for the quadratic function is:
f(x) = 4(x + 1)(x - 3)
To find an expression for the quadratic function given the x-intercepts and the range, we can start by using the fact that the x-intercepts are -1 and 3. The x-intercepts represent the solutions to the equation when the function equals zero. Therefore, the factors of the quadratic function are (x + 1) and (x - 3).
Next, since the range consists of all numbers less than or equal to 4, we can infer that the vertex of the quadratic function is the highest point on the graph, and it must have a y-coordinate of 4. The vertex has the form (h, k), where h represents the x-coordinate, and k represents the y-coordinate.
To find the vertex, we can use the formula h = -b/2a, where a, b, and c are coefficients of the quadratic function in the standard form ax^2 + bx + c. In this case, since we have the factors, we can rewrite the function as a(x + 1)(x - 3). Expanding this expression gives a(x^2 - 2x - 3).
Comparing this with the standard form, we see that a = 1. Now we can calculate the x-coordinate of the vertex:
h = -(-2) / (2 * 1) = 2
Since the y-coordinate of the vertex is 4, the vertex is (2, 4).
Now we can substitute the vertex values into the standard form of the quadratic function to find the value of c:
4 = a(2^2) + b(2) + c
4 = 4a + 2b + c
We also know that the factors of the quadratic function are (x + 1) and (x - 3), so when x = -1 or x = 3, the function equals zero. We can use these points to find equations that involve a, b, and c:
When x = -1:
0 = a(-1^2) + b(-1) + c
0 = a - b + c ............(1)
When x = 3:
0 = a(3^2) + b(3) + c
0 = 9a + 3b + c ............(2)
Now we have a system of three equations:
4 = 4a + 2b + c ......(3)
0 = a - b + c ..........(1)
0 = 9a + 3b + c ......(2)
Solving this system of equations will give us the values of a, b, and c, which we can substitute back into the standard form of the quadratic function to find the expression.
Incidentally, I can also provide the solution to the system of equations if you would like.