Math Problem Statement

y=-3x²-6x+4

Solution

The given equation is a quadratic function:

y=3x26x+4y = -3x^2 - 6x + 4

Key characteristics:

  1. Shape: Since the coefficient of x2x^2 is negative (-3), the parabola opens downward.
  2. Vertex: The vertex is the maximum point because the parabola opens downwards. To find the vertex, we use the formula for the xx-coordinate of the vertex:

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

Here, a=3a = -3, b=6b = -6. Substituting these values:

x=62(3)=66=1x = -\frac{-6}{2(-3)} = \frac{6}{-6} = -1

Now, substitute x=1x = -1 into the original equation to find the yy-coordinate:

y=3(1)26(1)+4=3(1)+6+4=7y = -3(-1)^2 - 6(-1) + 4 = -3(1) + 6 + 4 = 7

So, the vertex is (1,7)(-1, 7).

  1. Y-intercept: This is found by setting x=0x = 0:

y=3(0)26(0)+4=4y = -3(0)^2 - 6(0) + 4 = 4

So, the y-intercept is (0,4)(0, 4).

  1. Axis of Symmetry: The axis of symmetry is the vertical line through the vertex, so it's x=1x = -1.

  2. X-intercepts: To find the x-intercepts (where y=0y = 0), solve the quadratic equation:

3x26x+4=0-3x^2 - 6x + 4 = 0

We can use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=3a = -3, b=6b = -6, and c=4c = 4. Substituting these values:

x=(6)±(6)24(3)(4)2(3)=6±36+486=6±846=6±2216x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(-3)(4)}}{2(-3)} = \frac{6 \pm \sqrt{36 + 48}}{-6} = \frac{6 \pm \sqrt{84}}{-6} = \frac{6 \pm 2\sqrt{21}}{-6}

Simplifying:

x=1±213x = \frac{1 \pm \sqrt{21}}{3}

So the x-intercepts are:

x=1+213andx=1213x = \frac{1 + \sqrt{21}}{3} \quad \text{and} \quad x = \frac{1 - \sqrt{21}}{3}

Would you like more details on any of these steps or have any questions?

Related questions:

  1. How do you find the vertex of any quadratic equation?
  2. What is the significance of the axis of symmetry in a parabola?
  3. Can the quadratic formula always be used to find x-intercepts?
  4. How do transformations affect the graph of a quadratic function?
  5. What is the role of the discriminant in the quadratic formula?

Tip: Always double-check the discriminant (b24acb^2 - 4ac) to determine the number and nature of the roots before solving quadratic equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas

Formulas

Vertex formula: x = -b / 2a
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

Theorems

Quadratic Formula
Vertex Formula
Discriminant Theorem

Suitable Grade Level

Grades 9-11