Determine the Vertex and Axis of Symmetry for the parabola x=y2+4y+2x=y^2+4y+2, and determine which direction the parabola will open.

The parabola opens to the right; Vertex: (−2,−2)\left(-2,-2\right); Axis of Symmetry: y=−2y=-2

Determine the Vertex and Axis of Symmetry for the parabola x = y 2 + 4 y + 2 x=y^2+4y+2 , and determine which direction the parabola will open.

This problem asks us to determine the vertex and axis of symmetry for the parabola X equals Y2 + 4Y + 2. They were asked to determine which direction the parabola will open. OK. Now, right off the bat, what we should do is take a look at whatever this first term that we see is, this first variable. And I can see this starts with an X and then we have Y2 + 4, Y + 2 on the right side. And since we're starting with this X here, that tells me I'm dealing with a horizontal parabola, meaning it will either open to the right or left. Now, the standard form for a horizontal parabola looks like this. It's X is equal to A times Y minus K2 plus H. Now, is this what we have in our equation? Well, actually, no, we do not have this form. This is general form that we're given here, and then this is the standard form that we're trying to get. So because of that, what I'm going to need to do is complete the square to get our standard form. So, we have X is equal to y2 + 4 Y + 2. Now, to complete the square, what I need to do is take this middle term here, which I can see, which is 4. I need to divide it by 2 and then square it. So, I'm basically taking half of it and then squaring that value. Now, 4/2 is 2. That whole thing is going to be squared to give us 4. So what that means I need to add 4, but if I'm going to add 4, I also need to subtract 4 to keep this whole equation balanced. Now, this whole piece of the equation is going to factor into a perfect square, which is what we see right about there. So, it's going to factor into Y plus this value 2 squad, and then we have 2 minus 4, which just turns out to be -2. So, this is everything that X is equal to. So, now that we have our equation, this actually tells us everything we need to know since we're now in standard form. First off, I can see that we in essence, just have a positive 1 right here. I mean, it's not shown, but it's implicit because if we have nothing out front, that's the same as just having a positive 1 multiplied by this whole quantity. So, because it's a positive value, it's a horizontal parabola that will open to the right. As for the vertex, that's going to be at HK. Now, you can see that my H is clearly -2. And as for my K value, you'll notice that I'm adding 2 to K. I need to subtract the, or I'm adding 2 to Y, I should say, but I need to subtract a value from Y. So, because this is +2, we're going to actually write that as a -2. So, the vertex is gonna be at -2, -2, giving me my vertex. And then last, I need to find my X. of symmetry. The axis of symmetry for a horizontal parabola is going to be at Y equals K. So in this case, we're going to have Y equals my K value, which is -2, giving us our direction that the parabola opens, the vertex, and the axis of symmetry. Thus far we have solved this problem.

Determine the Vertex and Axis of Symmetry for the parabola x=y2+4y+2x=$$y^2+4y+2$$, and determine which direction the parabola will open.

The parabola opens to the right; Vertex: (−2,−2)\left(-2,-2\right); Axis of Symmetry: y=−2y=-2

The parabola opens upwards; Vertex: (−2,−2)\left(-2,-2\right)(−2,−2); Axis of Symmetry: y=−2y=-2y=−2

The parabola opens to the right; Vertex: (2,2)\left(2,2\right); Axis of Symmetry: y=−2y=-2y=−2

The parabola opens upwards; Vertex: (−2,−2)\left(-2,-2\right)(−2,−2); Axis of Symmetry: x=−2x=-2

Table of contents

Skip topic navigation

1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

2. Linear Equations and Inequalities3h 38m

The Addition and Subtraction Properties of Equality
41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Linear Inequalities in One Variable
1h 11m

3. Solving Word Problems2h 43m

Introduction to Problem Solving
37m

Formulas
21m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphing Linear Equations in Two Variables3h 17m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

5. Systems of Linear Equations1h 43m

Solving Systems of Linear Equations by Graphing
41m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

6. Exponents and Polynomials3h 25m

The Product Rule
10m

The Quotient Rule
13m

Intro to the Power Rules
18m

The Power of a Quotient Rule
18m

Negative Exponents
28m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
33m

Special Products
34m

7. Factoring2h 42m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
33m

Quadratic Equations & Applications
41m

8. Rational Expressions and Equations3h 13m

Simplifying Rational Expressions
39m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

9. Inequalities and Absolute Value2h 52m

Set Operations and Compound Inequalities
42m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

Linear Inequalities in Two Variables
28m

Systems of Linear Inequalities
23m

10. Relations and Functions2h 9m

Introduction to Relations and Functions
57m

Function Notation
15m

Graphing Common Functions
5m

Composition of Functions
24m

Direct & Inverse Variation
27m

11. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

12. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

13. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

14. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

15. Sequences, Series, and the Binomial Theorem1h 46m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
26m

Factorials
11m

14. Conic Sections & Systems of Nonlinear Equations

Parabolas

Beginning & Intermediate Algebra

14. Conic Sections & Systems of Nonlinear Equations

Parabolas

Previous problemNext problem

Struggling with Beginning & Intermediate Algebra?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Determine the Vertex and Axis of Symmetry for the parabola $x = y^{2} + 4 y + 2$ , and determine which direction the parabola will open.

The parabola opens upwards; Vertex: $$\left$(-2,-2$\right$)$ ; Axis of Symmetry: $y=-2$

The parabola opens to the right; Vertex: $open paren 2 comma 2 close paren$ ; Axis of Symmetry: $y=-2$

The parabola opens to the right; Vertex: $(- 2, - 2)$ ; Axis of Symmetry: $y = - 2$

The parabola opens upwards; Vertex: $$\left$(-2,-2$\right$)$ ; Axis of Symmetry: $x = - 2$

0 Comments

Previous problemNext problem

Verified step by step guidance

Rewrite the given equation $x = y^2 + 4y + 2$ to identify it as a parabola that opens horizontally (since $x$ is expressed in terms of $y$).

Complete the square for the $y$-terms on the right side to rewrite the equation in vertex form. Start by grouping the $y$ terms: $y^2 + 4y$.

Find the value to complete the square: take half of the coefficient of $y$ (which is 4), divide by 2 to get 2, then square it to get 4. Add and subtract 4 inside the equation to maintain equality.

Rewrite the equation as $x = (y + 2)^2 + (2 - 4)$, simplifying the constant terms to get the vertex form $x = (y + 2)^2 - 2$.

From the vertex form, identify the vertex as $(-2, -2)$ (since the equation is $x = (y - (-2))^2 - 2$), the axis of symmetry as the horizontal line $y = -2$, and determine that the parabola opens to the right because the squared term is positive and $x$ is expressed in terms of $y$.

4:49m

Watch next

Master Parabolas In Standard Form with a bite sized video explanation from Patrick Ford