Find the system of inequalities that forms the shaded area shown in the graph.
So in our graph, we can see that we have two functions. In using them, we have a shaded area. So, first we need to find the equation of these functions. But these functions aren’t necessarily equation; they’re inequalities. So they will have signs such as less than, greater than, less than or equal to, or greater than or equal to.
So let’s look at this shape. We have a solid curved line that’s U-shaped. And then in the blue, we have a dashed line that looks like a V. U-shaped graphs are usually quadratics, also known as parabolas. And their parent function is 𝑦 equals 𝑥 squared. V-shaped graphs are absolute value functions. And their parent function is 𝑦 equals the absolute value of 𝑥.
And when we say parent function, it just means the most simplified version. So these graphs would be at the origin. However, ours are not. They’re shifted to where their vertex, which is here, to the right three. And it’s also shifted up one. So we will have to look at a transformation of our functions. So in the red, we’ll be transforming the quadratic, the parabola. And in the blue, we will be transforming the absolute value graph.
So the 𝑎s deal with the steepness, like a slope. The ℎs deal with the horizontal shift and the 𝑘s with the vertical shift, so up and down. So let’s first look at our red graph. It was moved from the origin right three, which is a horizontal shift. So ℎ would be three. And then it’s moved up one. So 𝑘 would be one. Now with 𝑎, we need to check the rise of the graph. So from the vertex of our right graph, we need to check values.
So if we would go right one, that would be like plugging in one. One squared is one. So if we go right one, we should go up one, which we do. Now, what if we would go right two from our vertex? Well, two squared is four. So if we would go right two from our vertex, so here’s our vertex in blue, we go right two and up four. And this graph is symmetric so we don’t have to worry about the left as long as it matches, and it does.
And since it has the exact same values, 𝑎 would just be one. It didn’t change our graph. The steepness didn’t change. The values stayed the same, that we would have plugged in. So now we can plug these in. We have 𝑦 equals, there’s no need to write the one, so 𝑥 minus three squared plus one. Now, we have to remember that this actually won’t stay in equal sign. This isn’t an equality. So we will look at that at the end.
Now let’s find the equation for the absolute value graph. The absolute value graph shares this exact same vertex. So it went right three, which would be ℎ, and then it also went up one, which is 𝑘. Now let’s check our values. The absolute value of one is one. So if we start at our vertex in the green, if we would go right one, is our blue graph up one? No, it’s not. It’s actually up two. So somehow we need to get a two. Let’s check the next one.
If we go right two from the vertex, the absolute value of two is two. However, we actually go up four. So how did we go from one to two and two to four? What would we have multiplied by? It would have been two, so we need to make 𝑎 two because we’ve doubled our values. Instead of one, we have two, and that’s double. And instead of two, we got four, which is double. So 𝑎 is two, leaving us with 𝑦 equals two times the absolute value of 𝑥 minus three plus one.
So we are almost done. However, we need to change the equal sign to an inequality symbol. And this will do with the fact if the lines are solid or dashed in the shading. If we have a solid line, it must be less than or equal to or greater than or equal to. And if we have a dashed line, we must have less than or greater than. If we have shading above our graph, it must be greater than or greater than or equal to symbol. If we have shading below the graph, we must have a less than or less or equal to symbol.
So let’s begin with looking at our red graph, the parabola. It has a solid line and the shading is above it. So which symbol is shown up in the solid and the above? The greater than or equal to symbol. So this is the symbol we must use. Now for the blue, it’s a dashed line and the shading is actually below it. So which one they have in common? Less than! So we must use the less than symbol. Therefore, these two inequalities will form this system that forms the shaded area shown in the graph.