Describe how the graphs of y = |x| and y = |x + 3| are related.
Please help i am confused and explain if you can
when x is replaced by x+3, it means the graph shifts left by 3, since 3 has to be subtracted from the new values to equal the old values.
For instance, |2| = 2
|-1+3| = |2| = 2
You have to move left 3 so that x+3 is the same as the old x.
Ah, the dynamic duo of absolute values! Well, the graphs of y = |x| and y = |x + 3| are best buddies, but with a bit of a shift.
The graph of y = |x| is a classic V-shaped graph centered at the origin. It starts at (0, 0) and stretches upwards and downwards gracefully.
Now, the graph of y = |x + 3| is like the younger sibling of y = |x|. To make this graph, we take the original V-shaped graph, gently hold its hand, and move it 3 units to the left. So, the new vertex of the V will be at (-3, 0).
In other words, the second graph is just the first graph shifted to the left by 3 units.
To summarize, the two graphs have the same shape, but the second one is slightly to the left. It's like they're twins, with a minor difference that keeps them unique. Comedy and math, always a delightful combination!
The graphs of y = |x| and y = |x + 3| are related because they both represent absolute value functions. Here's a step-by-step explanation:
1. Starting with y = |x|, the graph of this function is a V-shaped curve that passes through the origin (0,0) and is symmetric with respect to the y-axis. The slope of the graph is positive for x > 0 and negative for x < 0.
2. Now, let's consider y = |x + 3|. This function is similar to y = |x|, but with a horizontal shift to the left by 3 units. This means that the entire graph of y = |x + 3| is shifted horizontally to the left by 3 units compared to the graph of y = |x|.
3. The new graph of y = |x + 3| still has the same V-shaped curve as y = |x|, but it is now centered at x = -3. This means that the graph passes through the point (-3,0) and is symmetric with respect to the vertical line x = -3. The slope of the graph is also positive for x > -3 and negative for x < -3.
In summary, the graphs of y = |x| and y = |x + 3| have similar shapes, but they are horizontally shifted relative to each other. The graph of y = |x + 3| is obtained by taking the graph of y = |x| and shifting it 3 units to the left.
To understand the relationship between the graphs of y = |x| and y = |x + 3|, let's break it down step by step:
1. Start with the graph of y = |x|: This is a V-shaped graph that opens upward and passes through the origin (0, 0). It is symmetric with respect to the y-axis, meaning that for every point (x, y) on the graph, the point (-x, y) is also on the graph.
2. Now, let's consider y = |x + 3|: This graph is similar to the graph of y = |x|, but it is shifted horizontally by 3 units to the left. In other words, every point on the graph of y = |x + 3| is 3 units to the left of the corresponding point on the graph of y = |x|.
To summarize, the graph of y = |x + 3| is the same shape as y = |x|, but it is shifted 3 units to the left. The vertical distance between the corresponding points on the two graphs remains the same, as the absolute value function preserves the distance from the x-axis.