Make predictions about how many times the graph of each equation below will touch the x axis. You may first want to rewrite some of the equations in a more useful form.=(x-2)(x-3).
A. y=(x+1)^2
B. y=x^2+6x+9
C. y=x^2+7x+10
D. y=x^2+6x+8
E. y=-x^2-4x-4
Thank you
You are welcome.
To predict how many times the graph of an equation will touch the x-axis, we can analyze the factors or characteristics of that equation.
Let's start by rewriting the equation given, (x - 2)(x - 3), in a more useful form.
(x - 2)(x - 3) can be expanded as x^2 - 5x + 6.
Now let's analyze each equation and make predictions:
A. y = (x + 1)^2
This equation is already in a useful form in the form of a perfect square. The exponent of 2 indicates that the graph will touch the x-axis at a single point. Therefore, the graph will touch the x-axis once.
B. y = x^2 + 6x + 9
This equation is also in a useful form, but it is another perfect square. The graph will touch the x-axis at a single point, just like in case A.
C. y = x^2 + 7x + 10
This equation is not a perfect square, but it can be factored as (x + 5)(x + 2). However, we can see that both factors are positive, which means that the parabola will remain above the x-axis throughout its curve. Therefore, it will not touch the x-axis.
D. y = x^2 + 6x + 8
By analyzing the equation, we see that it cannot be factored further. However, if we try to complete the square, we find that it is equivalent to (x + 3)^2 - 1. This means it's a perfect square, and the graph will touch the x-axis at a single point.
E. y = -x^2 - 4x - 4
This equation is also a perfect square, but it has a negative coefficient in front of the x^2 term. Therefore, the parabola will open downwards, and it will touch the x-axis at a single point.
To summarize the predictions, the graphs of equations A, B, D, and E will touch the x-axis once each, while the graph of equation C will not touch the x-axis.
A, Not really. That first parabola has its vertex right on the x axis at x = -1 so it only hits the axis there
B, = (x+3)(x+3), same deal, vertex on x axis at x = -3 so barely touches once
C. b^2-4ac = 49-40 so yes, hits twice
D. b^2-4ac = 36-32 yes again hits twice
(x+4)(x+2)
E. b^2-4ac = 16+16 so yes, twice again
(x-2)(x+2) at +2 and -2