Math Problem Statement

Given the function f(x) = (x + 3)(x - 5), find the zeros of the function and the vertex of the parabola.

Solution

The function given in the image is:

f(x)=(x+3)(x5)f(x) = (x + 3)(x - 5)

1) What are the zeros of the function?

To find the zeros of the function, set f(x)=0f(x) = 0 and solve for xx: (x+3)(x5)=0(x + 3)(x - 5) = 0 This gives us two solutions: x+3=0x=3x + 3 = 0 \quad \Rightarrow \quad x = -3 x5=0x=5x - 5 = 0 \quad \Rightarrow \quad x = 5

So, the zeros (or roots) of the function are:

  • Smaller x=3x = -3
  • Larger x=5x = 5

2) What is the vertex of the parabola?

Since the function is in factored form, we can find the x-coordinate of the vertex as the midpoint of the two zeros: xvertex=3+52=22=1x_{\text{vertex}} = \frac{-3 + 5}{2} = \frac{2}{2} = 1

To find the y-coordinate of the vertex, substitute x=1x = 1 into the function: f(1)=(1+3)(15)=4×(4)=16f(1) = (1 + 3)(1 - 5) = 4 \times (-4) = -16

Thus, the vertex of the parabola is: (1,16)(1, -16)


Would you like further clarification or additional details on any part of this? Below are some follow-up questions for deeper exploration:

  1. How would the graph of the function f(x)=(x+3)(x5)f(x) = (x+3)(x-5) look like?
  2. How can we rewrite the function f(x)f(x) in standard quadratic form?
  3. What is the axis of symmetry for this parabola?
  4. How can you find the y-intercept of this function?
  5. What does the negative value of the vertex's y-coordinate indicate about the graph?

Tip: The zeros of a quadratic function always help determine where the parabola intersects the x-axis.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas
Factored Form

Formulas

Zeros of quadratic function: Set each factor equal to zero
Midpoint formula: (x1 + x2) / 2
Evaluate function at x to find y-coordinate

Theorems

Roots of a quadratic equation
Vertex as midpoint of roots for parabolas

Suitable Grade Level

Grades 8-10