Math Problem Statement

Graph 𝑔(π‘₯) = π‘₯ 2 + 8π‘₯ + 12 using the vertex and finding the x-intercepts.

Solution

To graph the quadratic function g(x)=x2+8x+12g(x) = x^2 + 8x + 12, we can follow these steps:

1. Find the vertex

The vertex form of a quadratic equation is given by y=a(xβˆ’h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. We can find the vertex of a quadratic equation in standard form y=ax2+bx+cy = ax^2 + bx + c by using the formula:

x=βˆ’b2ax = \frac{-b}{2a}

For g(x)=x2+8x+12g(x) = x^2 + 8x + 12, the coefficients are:

  • a=1a = 1
  • b=8b = 8
  • c=12c = 12

Now, calculate the x-coordinate of the vertex:

x=βˆ’82Γ—1=βˆ’82=βˆ’4x = \frac{-8}{2 \times 1} = \frac{-8}{2} = -4

To find the y-coordinate of the vertex, substitute x=βˆ’4x = -4 into the original equation:

g(βˆ’4)=(βˆ’4)2+8(βˆ’4)+12=16βˆ’32+12=βˆ’4g(-4) = (-4)^2 + 8(-4) + 12 = 16 - 32 + 12 = -4

So, the vertex is at (βˆ’4,βˆ’4)(-4, -4).

2. Find the x-intercepts

To find the x-intercepts, set g(x)=0g(x) = 0 and solve for xx:

0=x2+8x+120 = x^2 + 8x + 12

Now, solve the quadratic equation using factoring:

x2+8x+12=(x+6)(x+2)=0x^2 + 8x + 12 = (x + 6)(x + 2) = 0

Set each factor equal to zero:

x+6=0orx+2=0x + 6 = 0 \quad \text{or} \quad x + 2 = 0

Solving these gives:

x=βˆ’6orx=βˆ’2x = -6 \quad \text{or} \quad x = -2

So, the x-intercepts are at x=βˆ’6x = -6 and x=βˆ’2x = -2.

3. Graph the function

Now that we know the following key points, we can sketch the graph:

  • The vertex is at (βˆ’4,βˆ’4)(-4, -4).
  • The x-intercepts are at x=βˆ’6x = -6 and x=βˆ’2x = -2.

Since the parabola opens upwards (because a=1a = 1, which is positive), we can plot the vertex and x-intercepts, and then sketch the curve that passes through these points.

Summary of Key Points:

  • Vertex: (βˆ’4,βˆ’4)(-4, -4)
  • x-intercepts: (βˆ’6,0)(-6, 0) and (βˆ’2,0)(-2, 0)
  • The parabola opens upwards.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing Parabolas

Formulas

Vertex formula: x = -b/(2a)
Factoring of quadratic expressions

Theorems

Vertex of a parabola
X-intercepts of a quadratic function

Suitable Grade Level

Grades 8-10