SOLUTION: It asks, without drawing the graph of the given equation, determine: How many x-intercepts the parabola has,wether its vertex lies above or below or on the x-axis. The problem is

Question 147536This question is from textbook algebra
: It asks, without drawing the graph of the given equation, determine: How many x-intercepts the parabola has,wether its vertex lies above or below or on the x-axis.
The problem is y=-x^2+2x-1. I have a hard time understanding how to solve it, please help me. Thank you=) This question is from textbook algebra

Found 2 solutions by jim_thompson5910, mangopeeler07:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
How many x-intercepts does the parabola have?

To find this out, we need to find out how many solutions there are. So we need to use the discriminant formula .

From we can see that , , and

Start with the discriminant formula

Plug in , , and

Square to get

Multiply to get

Subtract from to get

Since the discriminant is equal to zero, this means that there is only one real solution.

Since there is only one real solution, there is only one x-intercept. Because there is only one x-intercept, this means that the vertex must lie on the x-axis. However, this may not be so obvious. So let's find out where the vertex is.

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Where does the vertex lie?

To find out if the vertex is above or below the x-axis, we need to find the y-coordinate of the vertex. However, we first need to find the x-coordinate of the vertex.

To find the x-coordinate of the vertex, use this formula: .

Start with the given formula.

From , we can see that , , and .

Plug in and .

Multiply 2 and to get .

Divide.

So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .

Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.

Start with the given equation.

Plug in .

Square 1 to get 1.

Multiply 2 and 1 to get 2.

Combine like terms.

So the y-coordinate of the vertex is , which means that the the vertex is on the x-axis. So this confirms our original claim.

Answer by mangopeeler07(462) (Show Source): You can put this solution on YOUR website!
Here's an easy solution, step by step, no formulas involved: the x-intercept is whatever x is when y is zero. factor the original equation by taking out negative one and then factoring . Set it equal to zero. Then solve for each expression individually to equal zero. What minus 1 equals zero? You should have gotten 1 for both. That means the x-intercept is 1. There is only one x-intercept because you got the same solution for x in both expressions. Since there is only 1, it lies on the x-axis.

**Just in case you're wondering whether the parabola opens up or down, it opens down. Whenever is positive, it moves up, and vice versa. Think of it this way: If your is positive, then y wants to be positive too, so it moves upward. If your is negative, then y wants to be negative too, so it moves downward.**
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