The Vertex of the Parabola - Examples, Exercises and Solutions

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

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

• Step 1: Identify the coefficients $a = 1$ and $b = -6$.
• Step 2: Use the vertex formula $x = -\frac{b}{2a}$ 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 $f(x) = x^2 - 6x$ with coefficients $a = 1$ and $b = -6$.

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

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

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

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

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

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

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

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

$x = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3$

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

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

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

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

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).

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

By comparing, we have:

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

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

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

The given equation is $y = (x-1)^2 - 1$. This equation is in the vertex form, $y = a(x-h)^2 + k$, where $a$, $h$, and $k$ are constants.

In this case, the given equation can be written as $y = 1 \cdot (x-1)^2 + (-1)$, indicating that $a = 1$, $h = 1$, and $k = -1$.

The vertex form of the quadratic equation allows us to directly identify the vertex of the parabola as $(h, k)$.

From our identification, it is clear that the vertex $(h, k)$ of the parabola is $(1, -1)$.

Therefore, the vertex of the given parabola is $(1, -1)$.

To solve for the vertex of the parabola given by the equation $y = (x-3)^2 - 1$, we start by comparing the equation with the standard vertex form of a quadratic function: $y = a(x-h)^2 + k$.

In the given equation, $y = (x-3)^2 - 1$, we identify:
- $h = 3$, which corresponds to the horizontal shift of the parabola.
- $k = -1$, which represents the vertical shift.

Therefore, the vertex of the parabola is at the point $(h, k)$, which is $(3, -1)$.

Thus, the vertex of the parabola is $(3, -1)$.

To solve for the vertex of the parabola given by the equation $y = x^2 + 3$, we will follow these steps:

• Step 1: Identify the coefficients from the equation. We have $a = 1$, $b = 0$, and $c = 3$.
• Step 2: Calculate the x-coordinate of the vertex using the formula $x = -\frac{b}{2a}$. Since $b = 0$, the formula becomes $x = -\frac{0}{2 \times 1} = 0$.
• Step 3: Find the y-coordinate of the vertex by substituting $x = 0$ back into the equation. Calculate $y = (0)^2 + 3 = 3$.

Therefore, the vertex of the parabola is at the point $(0, 3)$.

This corresponds to choice 3: $(0,3)$.

To find the vertex of the parabola $y = x^2$, we follow these steps:

• Step 1: Identify the coefficients from the equation $y = x^2$. Here, $a = 1$, $b = 0$, and $c = 0$.
• Step 2: Use the vertex formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Substituting the values, we get:

Therefore, the x-coordinate of the vertex is $0$.

Step 3: Substitute $x = 0$ back into the equation to find the y-coordinate:

Thus, the y-coordinate of the vertex is also $0$.

Therefore, the vertex of the parabola $y = x^2$ is $(0,0)$.

The correct answer choice is: $(0,0)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the vertex formula to find $h$ and $k$.
• Step 3: Determine the coordinates of the vertex.

Now, let's work through each step:
Step 1: The given quadratic equation is $y = x^2 - 6$ where $a = 1$, $b = 0$, and $c = -6$.
Step 2: We use the vertex formula:

$h = -\frac{b}{2a}$
Substituting the values, $h = -\frac{0}{2 \cdot 1} = 0$.

$k = c - \frac{b^2}{4a}$
Using the given values, $k = -6 - \frac{0^2}{4 \cdot 1} = -6$.

Step 3: Therefore, the vertex of the parabola is at $(h, k) = (0, -6)$.

The solution to the problem is $(0, -6)$.

The given equation of the parabola is $y = (x+1)^2 - 1$.

This equation is already in the vertex form, $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

By comparing, we identify:
The expression $(x + 1)$ implies that $h = -1$ (since $x + 1$ is equivalent to $(x - (-1))$).
The constant $-1$ is the $k$ value.

Thus, the vertex $(h, k)$ is $(-1, -1)$.

Therefore, the vertex of the parabola is at the point $(-1,-1)$.

To solve this problem, let's identify the vertex of the given parabola in the form $y = (x - 3)^2$.

The parabola is already given in the vertex form of $y = (x - h)^2 + k$, which is a special case of the quadratic equation where the vertex ($h, k$) can be read directly from the equation.

• Step 1: We compare the given equation $y = (x - 3)^2$ with the standard vertex form $y = (x - h)^2 + k$.
• Step 2: Notice that $h = 3$ and since there is no additional term added or subtracted outside the squared term, $k = 0$.

Therefore, the vertex of the quadratic function $y = (x - 3)^2$ is $(3, 0)$.

The correct choice from the multiple-choice options provided is the one that matches $(3, 0)$.

The solution to the problem is the vertex $(3, 0)$.

To solve the problem of finding the vertex of the parabola $y = (x-3)^2 + 6$, we take the following steps:

Step 1: Identify the form of the given equation.
The equation is given in the vertex form of a quadratic function, which is generally expressed as $y = (x-h)^2 + k$.

Step 2: Recognize the coefficients.
In the given equation $y = (x-3)^2 + 6$, compare it with the standard form $y=(x-h)^2 + k$ to identify $h$ and $k$. Here, $h = 3$ and $k = 6$.

Step 3: Determine the vertex.
The vertex of the parabola, therefore, is directly given by the point $(h, k) = (3, 6)$.

As a conclusion, the vertex of the parabola described by the equation $y = (x-3)^2 + 6$ is located at the point $(3, 6)$.

The given equation is $y = (x-7) - 7$.

First, simplify the equation:
$y = x - 7 - 7 = x - 14$.

This is a linear equation in the form $y = x - 14$. However, since the problem asks about a vertex, there might be a reconsideration needed if a quadratic term is missing. Since the question specifies a parabola, let's convert back:

Let's convert this into a complete square form:

Express $y = (x - 7)^2 - 7$, assuming we meant to represent:
$y = (x-h)^2 + k$

The vertex form representation would look like:
$y = (x-7)^2 + (-7)$

Hence, the vertex is $(7, -7)$.

Therefore, the vertex of the parabola is $(7, -7)$.

To solve for the vertex of the parabola, we need to recognize that the equation $y = (x+8)^2 - 9$ is already in vertex form, which is $y = (x-h)^2 + k$.

Let's break down the given equation:

• Rewrite the given equation: $y = (x+8)^2 - 9$.
  This matches the vertex form $y = (x - h)^2 + k$.
• Identify values: Here, $h = -8$ and $k = -9$.

Thus, the vertex is located at the point $(-8, -9)$.

Comparing with the multiple-choice options provided:

• Choice 1: $(-8, 9)$ - Incorrect $k$.
• Choice 2: $(8, -9)$ - Incorrect $h$.
• Choice 3: $(8, 9)$ - Incorrect both $h$ and $k$.
• Choice 4: $(-8, -9)$ - Correct.

Therefore, the vertex of the parabola is $(-8, -9)$.