Parabola of the Form y=(x-p)²+k - Examples, Exercises and Solutions

To solve this problem, let us first note that the labeled point is $(0, -4)$, which suggests the parabola touches or intersects the y-axis at this point. Without more information indicating horizontal translation, it is reasonable to assume this is the vertex of the parabola, pointing down a simple transformation from $y=x^2$ to $y=x^2-4$.

Given the simplicity and symmetry (likely no $x$ coefficient subtracted or added), this directly translates to a parabola form with only a vertical shift downward.

Therefore, the algebraic representation of the given parabolic drawing is $y = x^2 - 4$.

The correct choice corresponding to this is $y = x^2 - 4$.

To solve this problem, follow these steps:

• Identify the point related to the parabola, which is given as $(5, 4)$.
• This point is likely the vertex of the parabola. The vertex form equation is $y = (x-h)^2 + k$.
• Substitute the vertex coordinates $(h, k) = (5, 4)$ into the vertex form.

Using these steps, substitute $h = 5$ and $k = 4$ into the vertex form:

$y = (x - 5)^2 + 4$

This matches the given point and reflects the parabola intersecting or having its vertex at (5, 4).

Therefore, the algebraic representation of the drawing is $y = (x-5)^2 + 4$.

To determine the algebraic representation, we use the vertex form of a parabola, which is $y = (x-h)^2 + k$. Here, the vertex is placed at $(-2, 7)$, thus plug these values into our equation: $h = -2$ and $k = 7$.

Consequently, the equation of the parabola becomes:

$y = (x + 2)^2 + 7$

This representation correctly describes a parabola that passes through the vertex at $(-2, 7)$ and opens upwards, as indicated by the absence of a negative sign or alternate coefficient in front of the square term.

Therefore, the correct choice corresponding to this problem formulation is:

$y = (x + 2)^2 + 7$

To solve this problem, the following steps are necessary:

We begin with the original function:

• $y = -x^2$

First, we apply the horizontal shift of 3 units to the left. Moving a graph left involves adding a number to $x$ in the equation. Hence, replace $x$ with $(x + 3)$. This manipulatively affects the original function:

$y = -(x + 3)^2$

Next, we apply the vertical shift of 4 units upward. This involves adding 4 to the function:

$y = -(x + 3)^2 + 4$

Therefore, the equation representing the parabola moved 3 spaces to the left and 4 spaces up is:

$y = -(x + 3)^2 + 4$

Verification against the choices confirms that the correct answer is choice (1):

• $y = -(x + 3)^2 + 4$

This is indeed the equation that results after applying the given transformations to the original function $y = -x^2$.

To solve this problem, we'll start by understanding the transformations required:

• The original function is $y = x^2$.
• We need to move this function 2 spaces to the right and 5 spaces upwards.

Step 1: Apply the horizontal shift 2 units to the right.
To shift a function horizontally, replace $x$ with $x - h$, where $h$ is the shift to the right. Thus, we replace $x$ with $x - 2$ to get:

$y = (x - 2)^2$.

Step 2: Apply the vertical shift 5 units upwards.
To shift a function vertically, add $k$ to the function, where $k$ is the number of units to shift up. Thus:

$y = (x - 2)^2 + 5$.

Combining these transformations, the equation of the transformed function is:

$y = (x - 2)^2 + 5$.

This matches the choice labeled as 3. Thus, the correct equation after translating the parabola 2 spaces to the right and 5 spaces upwards is:

$y = (x - 2)^2 + 5$.

To find the intersection of the function $y = (x+5)^2 + 2$ with the y-axis, we follow these steps:

• Step 1: Identify that the intersection with the y-axis occurs when $x = 0$.
• Step 2: Substitute $x = 0$ into the equation $y = (x+5)^2 + 2$.
• Step 3: Perform the substitution and simplify: $y = (0+5)^2 + 2$.
• Step 4: Simplify further: $y = 5^2 + 2 = 25 + 2 = 27$.

Thus, the intersection point with the y-axis is $(0, 27)$.

The correct answer is option 4: $(0, 27)$.

The problem asks us to find where the parabola given by $y = (x-3)^2 - 4$ intersects the y-axis. The intersection with the y-axis occurs where $x = 0$. Let's find the value of $y$ by substituting $x = 0$ into the equation:

$y = (0 - 3)^2 - 4$

Simplify inside the parentheses:

$y = (-3)^2 - 4$

Calculate $(-3)^2$:

$y = 9 - 4$

Subtract 4 from 9:

$y = 5$

Thus, the intersection of the function with the y-axis occurs at the point $(0, 5)$.

The correct answer from the choices provided is $(0,5)$.

To solve this problem, we need to apply a vertical shift to the function $y = (x-2)^2$.

When a function $y = f(x)$ is shifted vertically by a constant $k$, the new function becomes $y = f(x) + k$. In this problem, we need to shift the function three units up.

Given the original function $y = (x-2)^2$:

• Step 1: Identify the kind of transformation. We aim to move the function 3 spaces up.
• Step 2: Add 3 to the existing equation. This yields $y = (x-2)^2 + 3$.

The updated equation represents the translated parabola after shifting 3 units upwards.

Comparing this result with the given multiple-choice options, the correct corresponding equation is:

$y = (x-2)^2 + 3$.

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

• Step 1: Identify the given information
• Step 2: Recognize the formula for vertical translation
• Step 3: Apply the formula for a downward move
• Step 4: Compare to multiple-choice answers to find a match

Now, let's work through each step:
Step 1: The original equation is $y = (x-1)^2$.
Step 2: A translation "downward" results in the formula $y = (x-1)^2 + k$, with $k = -3$ for moving 3 units downward.
Step 3: Substitute $k = -3$ into the formula to get $y = (x-1)^2 - 3$.
Step 4: Among the choices provided, $y = (x-1)^2 - 3$ matches our formula for the translated equation.

Therefore, the solution to the problem is $y = (x-1)^2 - 3$, corresponding to choice 2.

To solve this problem, we must adjust the given function $y = x^2$ by moving it 10 units upwards. This transformation affects only the constant in the quadratic equation.

Let's consider the necessary steps:

• Step 1: Recognize that the original function is $y = x^2$.
• Step 2: Calculate the new function after moving 10 units upwards by adding 10 to the original formula. Thus, our new equation becomes $y = x^2 + 10$.
• Step 3: Confirm this result aligns with one of the provided answer choices. Reviewing the choices, we see:
  Choice 1: $y = x^2 + 10$
  Choice 2: $y = x^2 - 10$
  Choice 3: $y = x^2$
  Choice 4: $y = 10x^2$
• Step 4: Verify the correct choice is $y = x^2 + 10$, which corresponds to moving the original function 10 units up.

Therefore, the equation of the function moved 10 spaces up is $y = x^2 + 10$.

To solve this problem, we will follow these steps:

• Step 1: Identify the original function.
• Step 2: Apply the vertical shift of 5 units upward.
• Step 3: Write the resulting equation.

Let's go through each step:

Step 1: The given function is $y = -(x-3)^2 - 1$. This can be identified as a downward-facing parabola with its vertex at the point $(3, -1)$.

Step 2: To move the entire function 5 spaces up, we add 5 to the constant term $-1$ in the equation. The effect of this transformation is that the new vertex becomes $(3, -1 + 5) = (3, 4)$.

Step 3: Updating the function, we have:

$y = -(x-3)^2 - 1 + 5$

Simplify by combining the constants:

$y = -(x-3)^2 + 4$

This transformation results in the function moving 5 units up along the vertical axis to a new equation. The final equation is $y = -(x-3)^2 + 4$.

Therefore, the solution to the problem is $y=-(x-3)^2+4$, which is choice 4 from the given options.

To solve this problem, we will perform the following steps:

• Step 1: Identify the original function. It is $y = -(x-2)^2 + 4$.
• Step 2: Determine how many units to move the function. According to the problem, we move it 10 spaces down, which means we subtract 10 from the entire function.
• Step 3: Perform the vertical transformation by modifying the constant term. The new function is:

$y = -(x-2)^2 + 4 - 10$.

Step 4: Simplify the resulting expression:

$y = -(x-2)^2 - 6$.

This adjusted equation shows the original parabola moved 10 spaces down.

If we look at the given choices, our result corresponds to choice 3.

Therefore, the equation representing the function moved 10 spaces down is $y = -(x-2)^2 - 6$.

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

• Step 1: Identify the original function.
• Step 2: Apply the vertical transformation.
• Step 3: Confirm the new function equation matches a given choice.

Now, let's work through each step:
Step 1: The initial function is $y = -(x-6)^2$. This represents a parabola that opens downward, vertex at (6,0).
Step 2: To shift the graph of the function 4 units up, we add 4 to the entire function:
$y = -(x-6)^2 + 4$.
Step 3: Review the provided choices to find the match:
- Choice 4: $y=-(x-6)^2+4$.
This matches our transformation result.

Therefore, the solution to the problem is $y = -(x-6)^2 + 4$.

To determine the correct equation, we need to consider the vertex form of a parabola:

• The vertex form is given by $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex.
• In this problem, the vertex is $(4, 6)$.
• The equation becomes: $y = a(x-4)^2 + 6$.
• Since the parabola opens downwards, $a$ must be negative, implying $a = -1$.
• Thus, the equation is $y = -(x-4)^2 + 6$.

By comparing our derived expression with the options provided:

• $y=-(x-4)^2+6$ matches our derived equation.

Therefore, the correct equation is $y=-(x-4)^2+6$, corresponding to choice 3.

To solve for the algebraic representation of the parabola from the drawing:

• Step 1: Identify the vertex of the parabola. The drawing indicates the vertex at $(8, -2)$.
• Step 2: Write the vertex form of the parabola, $y = a(x - h)^2 + k$, using the vertex $(8, -2)$ as $h = 8$ and $k = -2$.
• Step 3: Determine the orientation of the parabola. The drawing suggests the parabola opens downward, indicating a negative value for $a$. Hence, $a < 0$.
• Step 4: Substitute the vertex and the orientation into the equation: $y = -1(x - 8)^2 - 2$, simplifying to $y = -(x - 8)^2 - 2$.

Therefore, the algebraic representation of the parabola is $y = -(x - 8)^2 - 2$.