SAT Tip: Lost in Translation
This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.
Harder SAT math questions might ask you how a graph or function changes when its equation is adjusted. For example, a line with the equation y = 4x + 2 will be two units higher on the y-axis than a graph with the equation y = 4x. On the SAT, you will need to identify whether the x-coordinate or the y-coordinate is changing, and use the rules in the table to correctly identify the new graph.
For example, let’s say we have the equation y = 6x + 2, and we wanted to know what the equation would be if the same line shifted three units to the left. “To the left” means we’re moving along the x-axis. Even though we’re heading towards the negative side, or decreasing along the x-axis, we’re actually going to add three units to the “x” to make that shift. The equation of y = 6x + 2 shifted three units to the left becomes y = 6(x + 3) + 2. That, in turn, becomes y = 6x + 18 + 2, or y = 6x + 20.
To double check, plug in y = 0 and see what the difference in the x-intercepts are:
In the original equation, when y = 0, x = -1/3. In the new equation, when y = 0, x = -10/3 or -3 1/3, indicating that the x-intercepts have indeed shifted three units to the left. These “shifts” describe movements that change the location of the graph, not its shape. These movements are called “translations.”
Let’s say we had a function described as f(x) = ax+b. What happens to the graph if we make “x” negative, so that the new function equation looks like f(x) = -ax+b?
Because this function essentially has the same form as y = mx + b we know it’s a linear function. Since only the x-coordinate is affected, this will change the tilt of the line (the slope), bringing it from positive to negative. Reversing the sign of the “x” creates a “mirror” image where the line is essentially reflected across the y-axis.
But what if “x” is multiplied or divided by a number other than 1 or -1? Then the slope of the equation will be changed even more, and the shape of what is graphed will change. Look for another article coming soon covering shifting parabolas and other advanced coordinate geometry shapes.
Remember, if you are ever in any doubt about the rules of translations, you can always pick values for x and plug them into the new function to double-check.
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