Find the vertex form of the function. Then find each of the following.
(A) Intercepts
(B) Vertex
(C) Maximum or minimum
(D) Range
s(x)= x^2-8x+7
To find the vertex form of the function, we can complete the square. The vertex form is given by: s(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
s(x) = x^2 - 8x + 7
s(x) = (x^2 - 8x) + 7
To complete the square, we need to add and subtract the square of half the coefficient of x:
s(x) = (x^2 - 8x + 16 - 16) + 7
s(x) = (x^2 - 8x + 16) - 16 + 7
s(x) = (x - 4)^2 - 9
Therefore, the vertex form of the function s(x) = x^2 - 8x + 7 is s(x) = (x - 4)^2 - 9.
(A) To find the x-intercepts, we set s(x) = 0 and solve for x:
(x - 4)^2 - 9 = 0
(x - 4)^2 = 9
Taking the square root of both sides:
x - 4 = ±√9
x - 4 = ±3
x = 4 ± 3
x = 7 or x = 1
Therefore, the x-intercepts are x = 7 and x = 1.
To find the y-intercept, we set x = 0:
s(0) = (0 - 4)^2 - 9
s(0) = (-4)^2 - 9
s(0) = 16 - 9
s(0) = 7
Therefore, the y-intercept is y = 7.
(B) The vertex of the parabola is given by (h, k) = (4, -9).
(C) Since the coefficient of the x^2 term is positive (1), the parabola opens upwards and the vertex represents the minimum point.
(D) The range of the function is all real numbers greater than or equal to -9, since the vertex is the minimum point. So, Range: [-9, ∞).
To find the vertex form of the function s(x) = x^2 - 8x + 7, we can complete the square.
Step 1: Rewrite the quadratic equation by grouping the x terms together:
s(x) = (x^2 - 8x) + 7
Step 2: Take half of the coefficient of x (-8 in this case), square it, and add it inside the parentheses to both sides of the equation:
s(x) = (x^2 - 8x + 16) + 7 - 16
Step 3: Simplify the equation inside the parentheses and combine like terms outside:
s(x) = (x - 4)^2 - 9
Now, let's find each of the following:
(A) Intercepts:
To find the x-intercepts, set s(x) equal to zero and solve for x:
0 = (x - 4)^2 - 9
(x - 4)^2 = 9
x - 4 = ±√9
x - 4 = ±3
Solving for x, we get two x-intercepts:
x = 4 + 3 = 7
x = 4 - 3 = 1
To find the y-intercept, substitute x = 0 into the equation:
s(0) = (0 - 4)^2 - 9
s(0) = 16 - 9
s(0) = 7
So, the y-intercept is 7.
(B) Vertex:
The vertex form of the quadratic equation is s(x) = (x - h)^2 + k, where the vertex is at the point (h, k).
In our equation, s(x) = (x - 4)^2 - 9, the vertex is (4, -9).
(C) Maximum or minimum:
Since the coefficient of the x^2 term is positive, the parabola opens upward, meaning it has a minimum value.
(D) Range:
The range of the function s(x) is all real numbers greater than or equal to the y-coordinate of the vertex, which is -9. Therefore, the range is (-∞, -9]
To find the vertex form of the function s(x), we need to complete the square. The general vertex form of a quadratic function is given by:
s(x) = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
First, let's rearrange the given equation s(x) = x^2 - 8x + 7:
s(x) = (x^2 - 8x) + 7
Next, we complete the square by adding and subtracting a term to the equation that satisfies the pattern (a + b)^2 = a^2 + 2ab + b^2. For s(x), the coefficient of the x term is -8, so we need to add (-8/2)^2 = 16 to complete the square:
s(x) = (x^2 - 8x + 16 - 16) + 7
Simplifying:
s(x) = (x^2 - 8x + 16) - 16 + 7
s(x) = (x - 4)^2 - 9
Now we have the vertex form of the function: s(x) = (x - 4)^2 - 9.
(A) To find the x-intercepts, we set s(x) = 0:
(x - 4)^2 - 9 = 0
(x - 4)^2 = 9
Taking the square root of both sides:
x - 4 = ±3
x = 4 ± 3
So the x-intercepts are x = 1 and x = 7.
(B) The vertex form of the function tells us that the vertex is given by (h, k). In this case, the vertex is (4, -9).
(C) Since the coefficient of the x^2 term is positive, the parabola opens upward, which means the vertex represents the minimum point of the graph. Therefore, the function has a minimum at the vertex.
(D) To find the range of the function, we need to consider the y-values. Since the parabola opens upward and the vertex is (-9), the range includes all values greater than or equal to -9, so the range is (-∞, -9].