Pregunta
Graph the parabola and find its vertex and the axis of the parabola.
Ask by Ross Delgado. in the United States
Jan 30,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The vertex of the parabola
is at
, and the axis of symmetry is
.
Solución
To graph the parabola
and determine its vertex and axis of symmetry, follow these steps:
1. Identify the Vertex and Axis of Symmetry
The general form of a quadratic function is:
For the given parabola:
a. Finding the Vertex:
The vertex
of a parabola in the form
can be found by completing the square.
Step-by-Step:
-
Start with the original equation:
-
Complete the square for the quadratic part:
- Take the coefficient of
, which is -6. - Take half of it:
. - Square it:
.
- Take the coefficient of
-
Rewrite the equation by adding and subtracting this square inside the equation:
-
Identify the vertex:So, the vertex is
.
b. Finding the Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Its equation is:
So, the axis of symmetry is
.
2. Graphing the Parabola
While I can’t provide a visual graph here, I can outline key points to help you plot the parabola:
Vertex:
Axis of Symmetry:
Finding Additional Points:
Choose x-values around the vertex and calculate the corresponding y-values.
-
For
: Point: -
For
: Point: -
For
: Point: -
For
: Point: -
For
: Point: -
For
: Point:
Plotting these points along with the vertex and drawing a symmetric U-shaped curve will give you the graph of the parabola.
3. Summary
- Vertex:
- Axis of Symmetry:
- Additional Points to Plot:
-
, , -
, ,
-
By plotting these points and ensuring symmetry around the axis
, you’ll accurately graph the parabola
.
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Extra Insights
To graph the parabola described by the equation
, we first complete the square. Rewriting the equation gives us
. Completing the square on
yields
. So, the equation can be expressed as
. This shows that the vertex of the parabola is at the point
and opens upward. The axis of symmetry is the vertical line
.
Another fun fact about parabolas is that they are not just mathematical constructs; they can be found in physics! The arc of a ball thrown in the air can be modeled by a parabolic equation like the one above. This means you could use these mathematical concepts to predict how high and how far your throw will go—just like a real-life athlete!
If you’re interested in pushing your understanding further, look into the quadratic formula and how it derives from completing the square. It’s a powerful tool not just for finding the roots of equations, but also for understanding the properties and graphing of quadratics. These insights can give you a fresh perspective on how math applies to various fields such as engineering and computer graphics.
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