Math Problem Statement
Sketch the graph of Y= -x^(2) +6x - 8
Solution
To sketch the graph of the function , we'll follow a similar step-by-step approach to analyze its key features, including the vertex, axis of symmetry, intercepts, and overall shape.
Step-by-Step Analysis
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Identify the Type of Function:
The function is a quadratic function. Its general form is , where , , and . -
Determine the Shape:
Since , the parabola opens downwards. -
Find the Vertex:
The vertex form of a parabola is given by . For this function: To find the y-coordinate of the vertex, substitute into the function: So, the vertex is at . -
Axis of Symmetry:
The axis of symmetry for a parabola is the vertical line passing through the vertex. In this case, it is . -
Find the Y-intercept:
The y-intercept occurs when : So, the y-intercept is at . -
Find the X-intercepts:
The x-intercepts occur when . We solve the quadratic equation: Multiply the equation by -1 to simplify: Factor the quadratic: So, and . The x-intercepts are at and . -
Sketch the Graph:
- Plot the vertex at .
- The parabola opens downwards.
- The axis of symmetry is the vertical line .
- Plot the y-intercept at .
- Plot the x-intercepts at and .
Graph of the Function
Based on the analysis above, the graph is a downward-opening parabola with the vertex at , x-intercepts at and , and a y-intercept at .
If you'd like a visual representation, here is a sketch of the graph:
y
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--+--------------------------- x
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Would you like more details on any step, or have any specific questions?
Related Questions:
- How would the graph change if the equation were ?
- What is the importance of finding the vertex in a quadratic function?
- How do you determine the direction in which a parabola opens?
- What is the role of the discriminant in finding the x-intercepts of a quadratic function?
- How can you convert a quadratic function from standard form to vertex form?
Tip: Knowing how to factor quadratic expressions can greatly simplify finding the x-intercepts of a quadratic function.
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Math Problem Analysis
Mathematical Concepts
Quadratic Equations
Vertex Form
Graphing
Formulas
Quadratic equation formula ax^2 + bx + c = 0
Vertex form formula x = -b/(2a)
Axis of symmetry formula x = -b/(2a)
Theorems
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Suitable Grade Level
Grades 9-10
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