Question

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)=x²−4x+3 s(x)= (Type your answer in vertex form.)

Answer 1

The quadratic function s(x) = x² - 4x + 3 can be written in vertex form as s(x) = (x - 2)² - 7. The vertex is (2, -7), which is also the point of the minimum, since the parabola opens upwards. The range of the function is y ≥ -7, as the function can have all y-values greater than or equal to -7.

Step-by-step explanation:

The student's question relates to converting a quadratic function into vertex form and then identifying various attributes of the function such as intercepts, vertex, maximum or minimum, and range. The given quadratic function is s(x) = x² - 4x + 3. To convert it to vertex form, we complete the square.

• Move the constant term to the right: x² - 4x = -3.
• Divide the coefficient of x by 2 and square it: (-4/2)² = 4.
• Add and subtract this square within the left side: x² - 4x + 4 - 4 = -3.
• Write the perfect square and simplify the right side: (x - 2)² - 4 - 3 = (x - 2)² - 7.

The vertex form of the function is therefore s(x) = (x - 2)² - 7. Now we can find the following attributes:

1. Intercepts: Set x and s(x) to zero and solve separately for x.
2. Vertex: Directly from the vertex form (h, k), where h = 2 and k = -7.
3. Maximum or Minimum: Since the quadratic opens upward (positive leading coefficient), it has a minimum at the vertex, k = -7.
4. Range: The range is all the y-values the function can take, which is y ≥ -7.