Vertex of Parabola Practice Problems & Formula Examples

Master finding parabola vertex using formulas and symmetry methods. Practice with step-by-step solutions and real quadratic function examples.

📚Master Parabola Vertex Finding Techniques
  • Apply the vertex formula x = -b/2a to find x-coordinate efficiently
  • Calculate vertex y-coordinate by substituting into original parabola equation
  • Use x-axis intersection points and symmetry to locate vertex position
  • Identify maximum and minimum points on upward and downward parabolas
  • Solve real quadratic functions like y = 5x² + 20x + 4 step-by-step
  • Determine when to use formula method versus intersection point method

Understanding The Vertex of the Parabola

Complete explanation with examples

Vertex of the Parabola

The vertex of the parabola indicates the highest or maximum point of a sad-faced parabola, and the lowest or minimum point of a happy-faced parabola.

The first method to find the vertex of the parabola: (with formula)

First step: We will find the XX of the vertex according to the formula x=(−b)2ax=\frac{(-b)}{2a}

Second step: We will place the XX of the vertex we have found into the original parabola equation to find the YY of the vertex.

Second method to find the vertex of the parabola: according to 2 points of intersection with the X-axis and use of symmetry

First step: Find two points of intersection of the parabola with the XX axis using the quadratic formula.

Second step: Find the XX of the vertex: the point that is exactly between two points of intersection. The calculation will be done through the average of two XXs of the intersection points.

Third step: Place the XX of the vertex we have found into the original parabola equation to solve for the YY of the vertex.

Detailed explanation

Practice The Vertex of the Parabola

Test your knowledge with 6 quizzes

Find the vertex of the parabola

\( y=(x-3)^2+6 \)

Examples with solutions for The Vertex of the Parabola

Step-by-step solutions included
Exercise #1

The following function has been plotted on the graph below:

f(x)=x2−8x+16 f(x)=x^2-8x+16

Calculate point C.

CCC

Step-by-Step Solution

To solve the exercise, first note that point C lies on the X-axis.

Therefore, to find it, we need to understand what is the X value when Y equals 0.

 

Let's set the equation equal to 0:

0=x²-8x+16

We'll use the preferred method (trinomial or quadratic formula) to find the X values, and we'll discover that

X=4

 

Answer:

(4,0) (4,0)

Video Solution
Exercise #2

The following function has been graphed below:

f(x)=x2−6x f(x)=x^2-6x

Calculate point C.

CCCAAABBB

Step-by-Step Solution

To solve this problem, we'll calculate the vertex of the parabola given by the quadratic function f(x)=x2−6x f(x) = x^2 - 6x .

  • Step 1: Identify the coefficients a=1 a = 1 and b=−6 b = -6 .
  • Step 2: Use the vertex formula x=−b2a x = -\frac{b}{2a} to find the x-coordinate of the vertex.
  • Step 3: Substitute the calculated x-coordinate back into the function to find the y-coordinate.

Now, let's compute:

Step 1: The function is f(x)=x2−6x f(x) = x^2 - 6x with coefficients a=1 a = 1 and b=−6 b = -6 .

Step 2: Apply the vertex formula: x=−−62×1=62=3 x = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3 .

Step 3: For x=3 x = 3 , substitute into f(x) f(x) to find the y-coordinate:

f(3)=(3)2−6×3=9−18=−9 f(3) = (3)^2 - 6 \times 3 = 9 - 18 = -9 .

Therefore, the coordinates of the point C, which is the vertex, are (3,−9)(3, -9).

The correct answer is (3,−9)(3, -9), which corresponds to the given correct choice.

Answer:

(3,−9) (3,-9)

Video Solution
Exercise #3

The following function has been graphed below.

f(x)=x2−6x+8 f(x)=x^2-6x+8

Calculate point B.

BBB

Step-by-Step Solution

To calculate point B, we should determine the vertex of the quadratic function f(x)=x2−6x+8 f(x) = x^2 - 6x + 8 .

The x-coordinate of the vertex can be found using the formula x=−b2a x = -\frac{b}{2a} .

In our equation, we have a=1 a = 1 and b=−6 b = -6 , therefore:

x=−−62×1=62=3 x = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3

Next, we substitute x=3 x = 3 back into the function to find the y-coordinate:

f(3)=32−6×3+8=9−18+8=−1 f(3) = 3^2 - 6 \times 3 + 8 = 9 - 18 + 8 = -1

Thus, the vertex, which is point B, is (3,−1) (3, -1) .

Therefore, the solution indicates that point B is at (3,−1) (3, -1) .

Answer:

(3,−1) (3,-1)

Video Solution
Exercise #4

The following function has been graphed below:

f(x)=−x2+5x+6 f(x)=-x^2+5x+6

Calculate point C.

BBBAAACCC

Step-by-Step Solution

To answer the question, we must first remember the formula for finding the vertex of a parabola:

Now let's substitute the known data into the formula:

-5/2(-1)=-5/-2=2.5

In other words, the x-coordinate of the vertex of the parabola is found when the X value equals 2.5.

Now let's substitute this into the parabola equation to find the Y value:

-(2.5)²+5*2.5+6= 12.25

Therefore, the coordinates of the vertex of the parabola are (2.5, 12.25).

Answer:

(212,1214) (2\frac{1}{2},12\frac{1}{4})

Video Solution
Exercise #5

Find the vertex of the parabola

y=(x+1)2 y=(x+1)^2

Step-by-Step Solution

The equation y=(x+1)2 y = (x+1)^2 is already in the vertex form y=(x−h)2+k y = (x-h)^2 + k , where (h,k)(h, k) is the vertex of the parabola.

By comparing, we have:

  • The expression inside the square is (x+1) (x+1) , which can be rewritten as (x−(−1)) (x - (-1)) . Thus, h=−1 h = -1 .
  • The term k k is not present, which means k=0 k = 0 .

Therefore, the vertex (h,k)(h, k) of the parabola is (−1,0)(-1, 0).

Thus, the correct answer is (−1,0)(-1, 0).

Answer:

(−1,0) (-1,0)

Video Solution

Frequently Asked Questions

Everything you need to know about The Vertex of the Parabola

What is the vertex formula for parabolas and how do you use it?

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The vertex formula is x = -b/2a, where a and b come from the standard form y = ax² + bx + c. First, substitute your values to find the x-coordinate, then plug this x-value back into the original equation to find the y-coordinate of the vertex.

How do you find the vertex of a parabola using two x-intercepts?

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This method involves three steps: 1) Find the x-intercepts by setting y = 0 and using the quadratic formula, 2) Calculate the average of these two x-values to get the vertex x-coordinate, 3) Substitute this x-value into the original equation to find the vertex y-coordinate.

What's the difference between maximum and minimum vertex points?

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When the coefficient 'a' is positive, the parabola opens upward (happy face) and the vertex represents the minimum point. When 'a' is negative, the parabola opens downward (sad face) and the vertex represents the maximum point.

When should I use the vertex formula versus the x-intercept method?

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Use the vertex formula x = -b/2a when you need a quick solution from the standard form. Use the x-intercept method when you already know the roots or when the quadratic factors easily, as this method leverages the parabola's symmetry.

How do you write vertex coordinates correctly?

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Vertex coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. For example, if x = -2 and y = -16, write the vertex as (-2, -16).

What are common mistakes when finding parabola vertices?

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Common errors include: forgetting the negative sign in x = -b/2a, mixing up the order of coordinates, assuming negative coordinates mean a downward parabola, and arithmetic mistakes when substituting back into the original equation.

Can a parabola vertex have negative coordinates?

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Yes, vertex coordinates can be negative, positive, or zero. The sign of the coordinates indicates the vertex position relative to the origin, not the parabola's orientation. The coefficient 'a' determines whether the parabola opens up or down.

How do you solve vertex problems step by step?

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Follow this process: 1) Identify coefficients a, b, and c from y = ax² + bx + c, 2) Calculate x-coordinate using x = -b/2a, 3) Substitute this x-value into the original equation to find y-coordinate, 4) Write the final answer as (x, y).

More The Vertex of the Parabola Questions

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The Vertex of the Parabola

Locate the Vertex of the Parabola: y = (x + 7)² + 15 Locate the Vertex: Solving y = (x-7) + 5 for the Quadratic Peak Find the Vertex: Analyzing the Parabola y = (x+6) + 6 Identify the Vertex in the Quadratic Equation: y = (x - 5) - 5 Determine the Vertex for the Parabola in Equation: y = (x-7)² - 7

Continue Your Math Journey

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