Translating Lines And Quadratic Equations A Comprehensive Guide

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Hey guys! 👋 Ever found yourself scratching your head over translations in coordinate geometry? Don't worry, you're not alone! Today, we're diving deep into understanding how translations work, especially when dealing with lines and quadratic equations. We'll tackle a couple of problems step by step, and by the end, you'll be a pro at this. Let's get started!

1. Translating the Line y = 4x - 6

The Question

So, the first question we're tackling involves a line with the equation y = 4x - 6. This line is being translated, and we need to figure out what the new equation (the bayangan garisnya, as the question puts it) looks like after the translation. We have two translations to consider:

  • Translation A: (-1, 3)
  • Translation B: (-3, -2)

And of course, we'll need to sketch the graphs to visualize what's happening. Ready? Let's break it down.

Understanding Translations

Before we jump into the math, let's quickly recap what a translation is. In simple terms, a translation is like sliding a shape (in our case, a line) to a new position without rotating or resizing it. Think of it as picking up the line and placing it somewhere else on the graph. The translation vector tells us how far to move the line horizontally (left or right) and vertically (up or down).

For a point (x, y), if we translate it by a vector (a, b), the new point (x', y') will be:

  • x' = x + a
  • y' = y + b

This concept is crucial for finding the equation of the translated line.

Step-by-Step Solution for Translation A (-1, 3)

Let's tackle the first translation. We have the line y = 4x - 6, and we're translating it by the vector (-1, 3). Here's how we can find the equation of the translated line:

  1. Consider a general point (x, y) on the original line. This point satisfies the equation y = 4x - 6. Remember, this equation defines the relationship between the x and y coordinates of any point on the line. So, no matter where you pick a point on this line, the y-coordinate will always be 4 times the x-coordinate, minus 6.

  2. Apply the translation. When we translate the point (x, y) by (-1, 3), we get a new point (x', y'). Using our translation rules:

    • x' = x + (-1) = x - 1
    • y' = y + 3

    These equations tell us how the coordinates of a point change when we apply the translation. If we know the original point (x, y), we can find the new point (x', y') using these formulas. Conversely, if we know the new point (x', y'), we can work backward to find the original point (x, y).

  3. Express x and y in terms of x' and y'. This is a key step. We want to rewrite our original equation in terms of the new coordinates (x', y'). To do this, we solve the equations from step 2 for x and y:

    • x = x' + 1
    • y = y' - 3

    Now we have expressions for x and y in terms of x' and y'. This is essential because we're going to substitute these expressions into the original equation of the line.

  4. Substitute into the original equation. Now comes the fun part! We substitute our expressions for x and y into the original equation y = 4x - 6:

    y' - 3 = 4(x' + 1) - 6

    This substitution replaces the original x and y with their translated counterparts, giving us an equation in terms of x' and y'. This new equation represents the translated line. All that's left to do now is simplify it.

  5. Simplify the equation. Let's simplify the equation to get it into a more familiar form:

    y' - 3 = 4x' + 4 - 6 y' - 3 = 4x' - 2 y' = 4x' + 1

    So, the equation of the translated line is y' = 4x' + 1. We've successfully translated the line! To make it look neater, we can simply replace x' with x and y' with y:

    y = 4x + 1

    This is the equation of the line after translation A. Notice that the slope (4) remains the same, but the y-intercept has changed.

Step-by-Step Solution for Translation B (-3, -2)

Now, let's tackle the second translation using the same method. We have the original line y = 4x - 6 and we're translating it by the vector (-3, -2). This translation will move the line further down and to the left compared to the first translation.

  1. Apply the translation. When we translate the point (x, y) by (-3, -2), we get a new point (x', y'). Using our translation rules:

    • x' = x + (-3) = x - 3
    • y' = y + (-2) = y - 2

    These equations describe how the coordinates change under translation B. Notice that the changes are different from translation A, reflecting the different translation vector.

  2. Express x and y in terms of x' and y'. Similar to before, we need to express x and y in terms of x' and y'. This will allow us to substitute into the original equation and find the equation of the translated line. Solving the equations from step 1, we get:

    • x = x' + 3
    • y = y' + 2

    These expressions are key to finding the translated equation. Make sure you understand how we derived them from the translation rules.

  3. Substitute into the original equation. Now, we substitute our expressions for x and y into the original equation y = 4x - 6:

    y' + 2 = 4(x' + 3) - 6

    This is the same substitution process we used for translation A, but with the new expressions for x and y. This step is crucial for finding the translated equation.

  4. Simplify the equation. Let's simplify the equation to get it into slope-intercept form (y = mx + b):

    y' + 2 = 4x' + 12 - 6 y' + 2 = 4x' + 6 y' = 4x' + 4

    So, the equation of the translated line is y' = 4x' + 4. Again, the slope remains the same, but the y-intercept has changed due to the translation. We can rewrite this as:

    y = 4x + 4

    This is the equation of the line after translation B. We've successfully translated the line using the second vector!

Graphing the Lines

To truly understand what's happening, let's visualize these translations. We'll need to graph three lines:

  1. The original line: y = 4x - 6
  2. The line after translation A: y = 4x + 1
  3. The line after translation B: y = 4x + 4
  • Original line (y = 4x - 6): To graph this, we can find two points. For example, when x = 0, y = -6, giving us the point (0, -6). When x = 1, y = -2, giving us the point (1, -2). Plot these points and draw a line through them.
  • Translated line A (y = 4x + 1): Similarly, when x = 0, y = 1, giving us the point (0, 1). When x = 1, y = 5, giving us the point (1, 5). Plot these points and draw a line.
  • Translated line B (y = 4x + 4): When x = 0, y = 4, giving us the point (0, 4). When x = -1, y = 0, giving us the point (-1, 0). Plot these points and draw a line.

If you graph these lines accurately, you'll see that the translated lines are parallel to the original line. This is because translations don't change the slope of a line; they only shift its position. The graphical representation really helps solidify the concept of translation.

2. Translating the Quadratic Equation y = x² + 3x + 2

The Question

Now, let's level up and tackle a quadratic equation! We have the equation y = x² + 3x + 2, and it's being translated by the vector T(-1, 2). We need to find the equation of the translated parabola and, as before, sketch the graphs to see what's going on.

Understanding Translations of Quadratic Equations

The basic principle of translation remains the same: we're shifting the graph without changing its shape or orientation. However, dealing with a parabola (the graph of a quadratic equation) is a bit more complex than dealing with a line. The key is still to apply the translation rules to a general point on the curve and then rewrite the equation in terms of the new coordinates.

Step-by-Step Solution

Let's break down the solution step by step:

  1. Consider a general point (x, y) on the original parabola. This point satisfies the equation y = x² + 3x + 2. Just like with the line, this equation defines the relationship between the x and y coordinates of any point on the parabola. The parabola's shape is determined by this specific relationship.

  2. Apply the translation. We're translating by the vector T(-1, 2). This means:

    • x' = x + (-1) = x - 1
    • y' = y + 2

    These equations tell us how the coordinates of a point on the parabola change when we apply the translation. The x-coordinate decreases by 1, and the y-coordinate increases by 2.

  3. Express x and y in terms of x' and y'. We need to solve the equations from step 2 for x and y:

    • x = x' + 1
    • y = y' - 2

    These expressions are crucial for rewriting the equation in terms of the new coordinates. They allow us to substitute the original x and y with their translated counterparts.

  4. Substitute into the original equation. This is where things get a little more algebraic. We substitute our expressions for x and y into the original equation y = x² + 3x + 2:

    y' - 2 = (x' + 1)² + 3(x' + 1) + 2

    This substitution replaces the original coordinates with the translated coordinates. The resulting equation represents the translated parabola. Now, we need to expand and simplify this equation.

  5. Simplify the equation. This step involves some algebraic manipulation. Let's expand and simplify:

    y' - 2 = (x'² + 2x' + 1) + 3x' + 3 + 2 y' - 2 = x'² + 2x' + 1 + 3x' + 3 + 2 y' - 2 = x'² + 5x' + 6 y' = x'² + 5x' + 8

    So, the equation of the translated parabola is y' = x'² + 5x' + 8. We've successfully translated the quadratic equation! We can rewrite this as:

    y = x² + 5x + 8

    This is the equation of the parabola after the translation. Notice how the coefficients have changed, reflecting the shift in the parabola's position.

Graphing the Parabolas

To visualize the translation, we'll graph both the original parabola and the translated parabola:

  1. The original parabola: y = x² + 3x + 2
  2. The translated parabola: y = x² + 5x + 8
  • Original parabola (y = x² + 3x + 2): To graph this, it helps to find the vertex, the axis of symmetry, and some key points. The x-coordinate of the vertex is given by -b/2a, where a = 1 and b = 3. So, x = -3/2. Plugging this into the equation, we find the y-coordinate of the vertex. We can also find the x-intercepts by setting y = 0 and solving the quadratic equation. Plot these points and sketch the parabola.
  • Translated parabola (y = x² + 5x + 8): Similarly, we find the vertex. The x-coordinate of the vertex is -5/2. Plugging this into the equation, we find the y-coordinate. Since the discriminant (b² - 4ac) is negative, this parabola doesn't have real roots (x-intercepts), meaning it doesn't cross the x-axis. Plot the vertex and some other points to sketch the parabola.

By graphing these parabolas, you'll see that the translated parabola is indeed a shifted version of the original parabola. The visual representation makes it clear how the translation affects the parabola's position on the coordinate plane.

Conclusion

And there you have it! We've walked through translating both a line and a quadratic equation. The key takeaway is understanding how translations shift the graph by changing the coordinates of each point. By applying the translation rules and substituting, we can find the equations of the translated figures. Remember, practice makes perfect, so try out some more examples to solidify your understanding. Keep up the great work, guys! 👍