Math Problem Statement

f(x) = 4x²-2x+12

Solution

The function given is:

f(x)=4x22x+12f(x) = 4x^2 - 2x + 12

Let's break down its key features:

1. Type of Function:

This is a quadratic function, which means its graph will be a parabola. Since the coefficient of x2x^2 is positive (4), the parabola opens upwards.

2. Vertex:

The vertex form of a quadratic function ax2+bx+cax^2 + bx + c can be found using the vertex formula:

x=b2ax = \frac{-b}{2a}

For f(x)=4x22x+12f(x) = 4x^2 - 2x + 12, we have:

  • a=4a = 4
  • b=2b = -2

So, the x-coordinate of the vertex is:

x=(2)2(4)=28=14x = \frac{-(-2)}{2(4)} = \frac{2}{8} = \frac{1}{4}

Now, plug this value back into the function to find the y-coordinate of the vertex:

f(14)=4(14)22(14)+12f\left(\frac{1}{4}\right) = 4\left(\frac{1}{4}\right)^2 - 2\left(\frac{1}{4}\right) + 12 =4(116)12+12= 4\left(\frac{1}{16}\right) - \frac{1}{2} + 12 =1412+12= \frac{1}{4} - \frac{1}{2} + 12 =124+12= \frac{1 - 2}{4} + 12 =14+12= -\frac{1}{4} + 12 =1+484= \frac{-1 + 48}{4} =474= \frac{47}{4}

So, the vertex is at:

(14,474)\left( \frac{1}{4}, \frac{47}{4} \right)

3. Axis of Symmetry:

The axis of symmetry is the vertical line that passes through the vertex, given by:

x=14x = \frac{1}{4}

4. Y-intercept:

To find the y-intercept, set x=0x = 0:

f(0)=4(0)22(0)+12=12f(0) = 4(0)^2 - 2(0) + 12 = 12

So, the y-intercept is at (0,12)(0, 12).

5. Range and Domain:

  • Domain: The domain of any quadratic function is all real numbers, so xRx \in \mathbb{R}.
  • Range: Since the parabola opens upwards and the vertex is the minimum point, the range is y474y \geq \frac{47}{4}.

Summary:

  • Vertex: (14,474)\left( \frac{1}{4}, \frac{47}{4} \right)
  • Axis of Symmetry: x=14x = \frac{1}{4}
  • Y-intercept: (0,12)(0, 12)
  • Domain: (,)(-\infty, \infty)
  • Range: [474,)\left[ \frac{47}{4}, \infty \right)

Would you like more details or have any questions?


Related Questions:

  1. How do you find the x-intercepts of f(x)=4x22x+12f(x) = 4x^2 - 2x + 12?
  2. What is the significance of the discriminant in a quadratic function?
  3. How can we convert f(x)f(x) into vertex form?
  4. What are the properties of the graph of f(x)=4x22x+12f(x) = 4x^2 - 2x + 12?
  5. How does changing the coefficient of x2x^2 affect the shape of the parabola?

Tip: The discriminant of a quadratic equation, b24acb^2 - 4ac, helps determine the nature of its roots (real or complex).

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas

Formulas

Vertex formula: x = -b / 2a
Quadratic function standard form: ax² + bx + c

Theorems

Properties of quadratic functions
Axis of symmetry
Vertex

Suitable Grade Level

Grades 9-10