Question

for each function f decide if the equation f(x)=0 has 0, 1, or 2 solutions. explain how you know in your workbook. a. graph b. graph c. graph d. graph e. graph f. graph a) eliga su respuesta... b) eliga su respuesta... c) eliga su respuesta... d) eliga su respuesta... e) eliga su respuesta... f) eliga su respuesta...

Explanation:

Part a:

Step1: Analyze the graph

The graph of function \( f(x) \) in part a is a parabola opening upwards. We check the number of times it intersects the \( x \)-axis (where \( y = 0 \)). From the graph, we can see that it intersects the \( x \)-axis at two distinct points.

Step2: Conclusion for part a

Since the graph intersects the \( x \)-axis at two points, the equation \( f(x)=0 \) has 2 solutions.

Part b:

Step1: Analyze the graph

The graph of function \( f(x) \) in part b is a parabola opening upwards and it touches the \( x \)-axis at exactly one point (the vertex lies on the \( x \)-axis).

Step2: Conclusion for part b

Since the graph touches the \( x \)-axis at one point (a repeated root), the equation \( f(x)=0 \) has 1 solution.

Part c:

Step1: Analyze the graph

The graph of function \( f(x) \) in part c is a parabola opening upwards. We check its intersection with the \( x \)-axis. The graph does not intersect the \( x \)-axis (it is above the \( x \)-axis for all \( x \) in its domain).

Step2: Conclusion for part c

Since the graph does not intersect the \( x \)-axis, the equation \( f(x)=0 \) has 0 solutions.

Part d:

Step1: Analyze the graph

The graph of function \( f(x) \) in part d is a parabola opening downwards. We check the number of times it intersects the \( x \)-axis. From the graph, we can see that it intersects the \( x \)-axis at two distinct points.

Step2: Conclusion for part d

Since the graph intersects the \( x \)-axis at two points, the equation \( f(x)=0 \) has 2 solutions.

Part e:

Step1: Analyze the graph

The graph of function \( f(x) \) in part e is a parabola opening downwards and it touches the \( x \)-axis at exactly one point (the vertex lies on the \( x \)-axis).

Step2: Conclusion for part e

Since the graph touches the \( x \)-axis at one point (a repeated root), the equation \( f(x)=0 \) has 1 solution.

Part f:

Step1: Analyze the graph

The graph of function \( f(x) \) in part f is a parabola opening downwards. We check its intersection with the \( x \)-axis. The graph does not intersect the \( x \)-axis (it is below the \( x \)-axis for all \( x \) in its domain? Wait, no, actually, looking at the graph, it is a downward - opening parabola and does not cross the \( x \)-axis. Wait, no, the graph is a downward - opening parabola and is below the \( x \)-axis? Wait, no, the vertex is at \( y=- 2\) or so? Wait, no, from the graph, the parabola is opening downwards and does not intersect the \( x \)-axis. Wait, no, let's re - check. The graph of part f: the parabola is opening downwards and its vertex is below the \( x \)-axis? No, wait, the vertex is at \( y = - 2\) (approx) and it does not cross the \( x \)-axis. Wait, no, actually, the graph of part f: the parabola is opening downwards and does not intersect the \( x \)-axis. Wait, no, maybe I made a mistake. Wait, the graph of part f: let's see, the parabola is opening downwards and is entirely below the \( x \)-axis? No, the \( x \)-axis is the horizontal line \( y = 0\). The graph of part f has its vertex at \( y=-2\) (or some negative value) and opens downwards? Wait, no, opening downwards means the coefficient of \( x^{2}\) is negative. If the vertex is at \( y=-2\) and it opens downwards, it will be below the \( x \)-axis and not intersect the \( x \)-axis. Wait, but maybe I misread. Wait, the graph of part f: the parabola is opening downwards and does not intersect the \( x \)-axis. So the equation \( f(x)=0\) has 0 solutions? Wait, no, wait the graph in part f: let's look again. The graph is a downward - opening parabola with vertex at \( y=-2\) (approx) and it does not cross the \( x \)-axis. So the number of solutions of \( f(x) = 0\) is 0? Wait, no, maybe I messed up. Wait, part e: the parabola is opening downwards and touches the \( x \)-axis at one point. Part f: the parabola is opening downwards and is below the \( x \)-axis, so no intersection. So for part f, the equation \( f(x)=0\) has 0 solutions.

Answer:

a. 2 solutions
b. 1 solution
c. 0 solutions
d. 2 solutions
e. 1 solution
f. 0 solutions