Final answer:
Starting with the linear equation y = 2x + 3 and point (2, 7), the translated equation after moving 4 units left and 5 units down is y = 2x + 11. The new coordinates of the translated point are (-2, 2).
Step-by-step explanation:
Translation of a Linear Function
Let's start with a linear equation, like y = 2x + 3. Now, let's define a point that lies on this function, for example, (2, 7). This point satisfies the equation because when we substitute x with 2, we get y = 2(2) + 3 = 7.
For the translated equation, we move 4 units to the left and 5 units down which involves adjusting the x and y values of all points on the line. The new equation becomes y - 5 = 2(x + 4) + 3. Simplifying this, we get the translated equation y = 2x + 11.
The new coordinates of our point after translation are found by subtracting 4 from the x-coordinate and 5 from the y-coordinate of the original point. So, the new point is (2 - 4, 7 - 5), which simplifies to (-2, 2).
Entering this data into a calculator or computer, we can verify that the new point indeed lies on the translated linear function. The translation has successfully moved the original point according to the given directions.