Math Problem Statement
y=-3x²-6x+4
Solution
The given equation is a quadratic function:
Key characteristics:
- Shape: Since the coefficient of is negative (-3), the parabola opens downward.
- Vertex: The vertex is the maximum point because the parabola opens downwards. To find the vertex, we use the formula for the -coordinate of the vertex:
Here, , . Substituting these values:
Now, substitute into the original equation to find the -coordinate:
So, the vertex is .
- Y-intercept: This is found by setting :
So, the y-intercept is .
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Axis of Symmetry: The axis of symmetry is the vertical line through the vertex, so it's .
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X-intercepts: To find the x-intercepts (where ), solve the quadratic equation:
We can use the quadratic formula:
where , , and . Substituting these values:
Simplifying:
So the x-intercepts are:
Would you like more details on any of these steps or have any questions?
Related questions:
- How do you find the vertex of any quadratic equation?
- What is the significance of the axis of symmetry in a parabola?
- Can the quadratic formula always be used to find x-intercepts?
- How do transformations affect the graph of a quadratic function?
- What is the role of the discriminant in the quadratic formula?
Tip: Always double-check the discriminant () 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
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