Sketch a graph y = |x – 3| – 2 and describe the translations.
I don't understand this at all
Start with y = | x |
you should be able to graph this
if not, look at Wolfram:
http://www.wolframalpha.com/input/?i=y+%3D+%7Cx%7C
a big V with nothing below the x-axis
consider: y = |x| - 2 , we just moved the entire graph down 2 units.
http://www.wolframalpha.com/input/?i=y+%3D+%7Cx%7C+-+2
Now we will do a horizontal shift of 3 units to the right
y = |x-3| - 2
http://www.wolframalpha.com/input/?i=y+%3D+%7Cx-3%7C+-+2
or all in 1 graph:
http://www.wolframalpha.com/input/?i=plot+y+%3D+%7Cx%7C,+y+%3D+%7Cx%7C+-+2+,+y+%3D+%7Cx-3%7C+-+2
in general, if
y = f(x)
then f(x + a) will move f(x) a units to the left
then f(x - a) will move f(x) a units to the right
then f(x) + b , will move f(x) b units up
then f(x) - b , will move f(x) b units down
can you see how I applied these rules to your equation ?
Alright, let me try to put it in simple terms with a dash of humor! Imagine you're playing a game of hide and seek with your graph. The equation y = |x – 3| – 2 is like telling the graph where to hide and how to disguise itself.
First, let's focus on "x – 3". Think of it as the graph's way of saying, "I wanna hide 3 units to the right!" So, the poor graph uncomfortably scooches to the right by 3 units.
Next, the absolute value sign | | appears and tells the graph that it can't hide in the negative realm. So the graph is forced to flip itself to the positive side of the y-axis. It's like saying, "No hiding in the negativity zone!"
Lastly, the "- 2" comes into play. It's like the graph is saying, "I'm tired of being too high up! I need to come down a bit!" So it subtracts 2 units from its height and slides down accordingly.
And there you have it! The graph y = |x – 3| – 2 has made its translations, hiding to the right by 3 units, flipping to the positive side, and lowering itself by 2 units. Voila!
To sketch the graph of y = |x – 3| – 2, we can break it down into steps and describe the translations:
Step 1: Start with the graph of y = |x|, which is a V-shaped graph passing through the origin and opening upwards.
Step 2: Perform a translation 3 units to the right. This means every x-coordinate in the original graph will increase by 3. So, the vertex of the V-shape will now be at (3, 0).
Step 3: Perform a translation 2 units downwards. This means every y-coordinate in the previous step's graph will decrease by 2. So, the entire graph is shifted down by 2 units.
Combining these translations, the graph y = |x – 3| – 2 will have its vertex at (3, -2), and it will still have a V-shape opening upwards but slightly shifted to the right and down.
No problem! Let's break it down step by step.
First, let's understand the equation. The graph is described by the equation y = |x – 3| – 2. The |x – 3| part represents the absolute value of the expression (x – 3). Absolute value ensures that the output is always positive or zero, regardless of the input's sign. Lastly, subtracting 2 from the absolute value output shifts the entire graph down by 2 units.
To sketch the graph, follow these steps:
1. Identify the base function: In this case, it is y = |x|.
2. Apply the necessary horizontal shift: The base function has its vertex at (0, 0). The expression (x – 3) shifts the entire graph horizontally by 3 units to the right. So, mark a point at (3, 0).
3. Plot additional points: Now, consider a few other values for x, like x = 0, -1, -2, and 1. Substitute these values into the equation to get y-values. For example, when x = 0, y is calculated as |0 – 3| – 2 = |-3| – 2 = 3 – 2 = 1. So, plot the point (0, 1).
4. Reflect and plot remaining points: Since we have an absolute value function, the graph is symmetric with respect to the y-axis. So, plot the mirror image of the points you've already plotted across the y-axis.
5. Connect the points: Use a smooth curve to connect the plotted points.
The resulting graph will be a V-shaped graph, opening upwards, with the vertex at (3, -2).
Note: If you're struggling with visualization, you can also use graphing software or online graphing tools to plot the equation and generate the graph for you.
Use only the positive(absolute) value of the results inside the brackets. The student should do P5 and P6, and use the six points for graphing. The sketch will resemble a parabola that opens upward.
Y = |x-3| - 2.
Y = |0-3| - 2 = 3 - 2 = 1. P1(0,1).
Y = |1-3| - 2 = 2 - 2 = 0. P2(1,0).
Y = |2-3| - 2 = 1 - 2 = -1. P3(2,-1).
Y = |3-3| - 2 = 0 - 2 = -2. P4(3,-2).
Y = |4-3| - 2 = ? - 2 = ? P5(4,?).
Y = |5-3| - 2 = ? - 2 = ? P6(?, ?).