Math Problem Statement

Given the quadratic equation in vertex form y = (x + 1)^2 - 4, determine the vertex, standard form, y-intercept, intercept form, and x-intercepts, and then graph the parabola.

Solution

Let's solve each part step by step for the quadratic equation provided:

The given equation is:

y=(x+1)24y = (x + 1)^2 - 4


(A) What is the vertex of the parabola?

The equation is in vertex form: y=a(xh)2+ky = a(x-h)^2 + k, where (h,k)(h, k) is the vertex.

Here, (x+1)2(x + 1)^2 can be written as (x(1))2(x - (-1))^2, so:

  • h=1h = -1
  • k=4k = -4

Thus, the vertex is:
(1,4)(-1, -4)


(B) What is the equation in standard form?

To convert from vertex form to standard form y=ax2+bx+cy = ax^2 + bx + c, expand (x+1)2(x + 1)^2:

y=(x+1)24y = (x + 1)^2 - 4 y=x2+2x+14y = x^2 + 2x + 1 - 4 y=x2+2x3y = x^2 + 2x - 3

The standard form is:
y=x2+2x3y = x^2 + 2x - 3


(C) What is the yy-intercept of the equation?

The yy-intercept occurs when x=0x = 0. Substitute x=0x = 0 into the standard form y=x2+2x3y = x^2 + 2x - 3:

y=(0)2+2(0)3y = (0)^2 + 2(0) - 3 y=3y = -3

Thus, the yy-intercept is:
(0,3)(0, -3)


(D) What is the equation in intercept form?

Intercept form is y=a(xm)(xn)y = a(x - m)(x - n), where mm and nn are the xx-intercepts.

From standard form y=x2+2x3y = x^2 + 2x - 3, we factorize:

x2+2x3=(x+3)(x1)x^2 + 2x - 3 = (x + 3)(x - 1)

Thus, the equation in intercept form is:
y=(x+3)(x1)y = (x + 3)(x - 1)


(E) What are the xx-intercepts?

The xx-intercepts occur when y=0y = 0. From the intercept form:

(x+3)(x1)=0(x + 3)(x - 1) = 0

Setting each factor equal to zero:

  1. x+3=0x=3x + 3 = 0 \Rightarrow x = -3
  2. x1=0x=1x - 1 = 0 \Rightarrow x = 1

Thus, the xx-intercepts are:
(3,0) and (1,0)(-3, 0) \text{ and } (1, 0)


(F) Graph the parabola

Use the points found:

  1. Vertex: (1,4)(-1, -4)
  2. Y-intercept: (0,3)(0, -3)
  3. X-intercepts: (3,0)(-3, 0) and (1,0)(1, 0)

You can plot these points on the provided graph to sketch the parabola.


Summary of Answers:

  • (A): Vertex: (1,4)(-1, -4)
  • (B): Standard form: y=x2+2x3y = x^2 + 2x - 3
  • (C): Y-intercept: (0,3)(0, -3)
  • (D): Intercept form: y=(x+3)(x1)y = (x + 3)(x - 1)
  • (E): X-intercepts: (3,0)(-3, 0) and (1,0)(1, 0)

Would you like me to graph this parabola for you or explain any of these steps in more detail?


Related Questions:

  1. How do you derive the vertex form of a quadratic equation from standard form?
  2. What is the significance of the vertex in a parabola?
  3. How do you calculate the axis of symmetry for a parabola?
  4. Can a parabola have no x-intercepts? Why?
  5. How does the coefficient aa in y=ax2+bx+cy = ax^2 + bx + c affect the shape of the parabola?

Tip: When working with quadratic equations, always check the discriminant b24acb^2 - 4ac to determine the number of x-intercepts.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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