Math Problem Statement

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Part 1

How can you find the maximum or minimum value of a quadratic​ function?

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Part 1

Choose the correct answer below.

A.

Evaluate the function with

xequals=0.

The resulting​ y-value is the minimum or maximum value depending on whether the graph opens upward or downward.

B.

Find the vertex. The​ y-coordinate of the vertex is the minimum or maximum value depending on whether the graph opens upward or downward.

C.

Find the​ y-intercept. The​ y-intercept is the minimum or maximum value depending on whether the graph opens upward or downward.

D.

Find the vertex. The​ x-coordinate of the vertex is the minimum or maximum value depending on whether the graph opens upward or downward.

Solution

The correct answer is:

B. Find the vertex. The y-coordinate of the vertex is the minimum or maximum value depending on whether the graph opens upward or downward.

Explanation:

In the case of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c, the maximum or minimum value occurs at the vertex of the parabola. The y-coordinate of the vertex represents this value:

  • If a>0a > 0 (the parabola opens upwards), the vertex gives the minimum value.
  • If a<0a < 0 (the parabola opens downwards), the vertex gives the maximum value.

Would you like further explanation on this or have more questions?

Here are five related questions for practice:

  1. How do you find the x-coordinate of the vertex using the formula x=b2ax = -\frac{b}{2a}?
  2. What happens to the vertex if the quadratic function is shifted vertically?
  3. Can the vertex ever be the x-intercept of the quadratic function?
  4. How does the value of aa affect whether the vertex is a maximum or minimum?
  5. How do you find the axis of symmetry from a quadratic function?

Tip: In the vertex form of a quadratic function, f(x)=a(xh)2+kf(x) = a(x - h)^2 + k, the vertex is directly given as (h,k)(h, k), and the y-coordinate kk is the maximum or minimum value.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex of Parabola

Formulas

Vertex formula x = -b / (2a)
Quadratic function f(x) = ax^2 + bx + c

Theorems

Vertex theorem for parabolas

Suitable Grade Level

Grades 9-10