Math Problem Statement
Solution
Let's solve each part step by step for the quadratic equation provided:
The given equation is:
(A) What is the vertex of the parabola?
The equation is in vertex form: , where is the vertex.
Here, can be written as , so:
Thus, the vertex is:
(B) What is the equation in standard form?
To convert from vertex form to standard form , expand :
The standard form is:
(C) What is the -intercept of the equation?
The -intercept occurs when . Substitute into the standard form :
Thus, the -intercept is:
(D) What is the equation in intercept form?
Intercept form is , where and are the -intercepts.
From standard form , we factorize:
Thus, the equation in intercept form is:
(E) What are the -intercepts?
The -intercepts occur when . From the intercept form:
Setting each factor equal to zero:
Thus, the -intercepts are:
(F) Graph the parabola
Use the points found:
- Vertex:
- Y-intercept:
- X-intercepts: and
You can plot these points on the provided graph to sketch the parabola.
Summary of Answers:
- (A): Vertex:
- (B): Standard form:
- (C): Y-intercept:
- (D): Intercept form:
- (E): X-intercepts: and
Would you like me to graph this parabola for you or explain any of these steps in more detail?
Related Questions:
- How do you derive the vertex form of a quadratic equation from standard form?
- What is the significance of the vertex in a parabola?
- How do you calculate the axis of symmetry for a parabola?
- Can a parabola have no x-intercepts? Why?
- How does the coefficient in affect the shape of the parabola?
Tip: When working with quadratic equations, always check the discriminant to determine the number of x-intercepts.
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Math Problem Analysis
Mathematical Concepts
Quadratic Equations
Vertex Form
Standard Form
Intercept Form
Graphing Parabolas
Formulas
Vertex Form: y = a(x - h)^2 + k
Standard Form: y = ax^2 + bx + c
Factoring Quadratics: y = a(x - m)(x - n)
Theorems
Zero Product Property
Suitable Grade Level
Grades 8-10
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