Challenge #3 - Horizonal and Vertical Shifts
The graph shows five sets of points in five different colors. Set #1 includes red points at located at ( negative 1 , 1 ), ( 0 , 0 ), and ( 1 , 1 ). There is a red parabola passing through these points. Set #2 includes blue points that are shifted three units up from the red points. Set #3 includes green points that are shifted four units to the right of the red points. Set #4 includes orange points that are shifted three units down from the red points. And finally, Set #5 includes purple points that are shifted four units to the left of the red points.The red parabola passes through the red points.The blue parabola does not pass through the blue points. Please try again.The green parabola does not pass through the green points. Please try again.The orange parabola does not pass through the orange points. Please try again.The purple parabola does not pass through the purple points. Please try again.
Plot five parabolas, one through each set of color-coordinated points. One is done for you.



Press "Check My Work" to see your results.

Parabola
Equation
Red
Blue
Green
Orange
Purple

To solve this problem, we need to determine the equations for the parabolas based on the given sets of points and the transformations applied to the red parabola.

### Step 1: Determine the equation of the red parabola.
The red parabola passes through the points (-1, 1), (0, 0), and (1, 1). These points suggest that the red parabola is centered at the origin, and it seems to follow a standard form of a quadratic function:
\[
y = x^2
\]
This is the equation of the red parabola, which passes through the points (-1, 1), (0, 0), and (1, 1).

### Step 2: Apply transformations to the red parabola to obtain the other parabolas.
Each of the other sets of points is formed by shifting the red points either vertically or horizontally. For a function \(y = f(x)\), the transformations are as follows:
- **Vertical shift**: \(y = f(x) + k\) (shifts the graph up by \(k\) units if \(k > 0\) or down by \(k\) units if \(k < 0\)).
- **Horizontal shift**: \(y = f(x - h)\) (shifts the graph right by \(h\) units if \(h > 0\) or left by \(h\) units if \(h < 0\)).

### Step 3: Find the equations for each parabola.

#### 1. **Blue Parabola** (Shifted 3 units up)
This set of points is shifted 3 units up from the red points, so the equation for the blue parabola is:
\[
y = x^2 + 3
\]

#### 2. **Green Parabola** (Shifted 4 units to the right)
This set of points is shifted 4 units to the right from the red points, so the equation for the green parabola is:
\[
y = (x - 4)^2
\]

#### 3. **Orange Parabola** (Shifted 3 units down)
This set of points is shifted 3 units down from the red points, so the equation for the orange parabola is:
\[
y = x^2 - 3
\]

#### 4. **Purple Parabola** (Shifted 4 units to the left)
This set of points is shifted 4 units to the left from the red points, so the equation for the purple parabola is:
\[
y = (x + 4)^2
\]

### Final Equations:
- Red parabola: \(y = x^2\)
- Blue parabola: \(y = x^2 + 3\)
- Green parabola: \(y = (x - 4)^2\)
- Orange parabola: \(y = x^2 - 3\)
- Purple parabola: \(y = (x + 4)^2\)

These equations describe the five parabolas, each passing through the respective points after the specified transformations.

Would you like more details or have any questions about this solution? Here are 5 related questions for further exploration:

1. How does the equation of a parabola change when it is reflected across the x-axis?
2. What is the effect of shifting a quadratic function horizontally by \(h\) units?
3. Can you find the vertex of the purple parabola \(y = (x + 4)^2\)?
4. How would the equation change if we shifted the red parabola both vertically and horizontally?
5. What is the general form of a quadratic function, and how do transformations affect it?

**Tip**: When working with transformations, remember that horizontal shifts are applied inside the function, while vertical shifts are applied outside the function.

Challenge #3 - Horizontal and Vertical Shifts. The graph shows five sets of points in five different colors. Set #1 includes red points located at (-1, 1), (0, 0), and (1, 1). There is a red parabola passing through these points. Set #2 includes blue points that are shifted three units up from the red points. Set #3 includes green points that are shifted four units to the right of the red points. Set #4 includes orange points that are shifted three units down from the red points. And finally, Set #5 includes purple points that are shifted four units to the left of the red points. The red parabola passes through the red points. The blue parabola does not pass through the blue points. Please try again. The green parabola does not pass through the green points. Please try again. The orange parabola does not pass through the orange points. Please try again. The purple parabola does not pass through the purple points. Please try again. Plot five parabolas, one through each set of color-coordinated points.

Learn how to solve problems involving the horizontal and vertical shifts of parabolas. This tutorial guides you through plotting five parabolas based on transformations of the equation y = x^2. Explore key concepts such as vertical and horizontal shifts, quadratic equations, and transformations. Suitable for students in Grades 9-11.

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