The -intercept of the graph of , a parabola, can be found by setting  and solving for :

The -intercept is .

The -intercepts can be found by  and solving for :

, in which case , or

, in which case . 

The two -intercepts are  and .

The triangle is shown below:

If the segment connecting the points on the -axes is taken as the base, then 

The height is therefore the vertical distance from this segment to the point on the -axis, which is 

The area is half their product, or

.

I) The orientation of the parabola is determined solely by the value of . Since , the parabola can be determined to open upward.

II and V) The -coordinate of the vertex is ; since you are not given , you cannot find this. Also, since the line of symmetry has equation , for the same reason, you cannot find this either.

III) The -intercept is the point at which ; by substitution, it can be found to be at .  is unknown, so the -intercept cannot be found.

IV) The -intercept(s), if any, are the point(s) at which . This is solvable using the quadratic formula

Since all three of  and  must be known for this to be evaluated, and only  is known, the -intercept(s) cannot be identified.

The correct response is I only.

The eqiatopm of  is given in the vertex form

,

so the vertex of its parabola is . The graphs of  and  are parabolas with the same vertex, so they must have the same values for  and . 

For the function ,  and .

Of the five choices, the only equation of   that has these same values, and that therefore has a parabola with the same vertex, is .

To verify, graph both functions on the same grid.

Let  and  be the - and -intercepts, respectively, of the line. Then the slope of the line is , or, equilvalently, . 

We do not need to find the actual slopes of the four choices if we observe that in each case,  and  are of the same sign. Since the quotient of two numbers of the same sign is positive, it follows that  is negative, and therefore, none of the pairs of intercepts can be those of a line with positive slope .

 Let  and  be the - and -intercepts, respectively, of the line. Then the slope of the line is , or, equilvalently, . 

We can examine the intercepts in each choice to determine which set meets these conditions.

 and :

Slope: 

 and 

Slope: 

 and 

Slope: 

 and 

Slope: 

 and  comprise the correct choice.

The slope of the green line can be calculated by noting that the - and -intercepts of the line are, respectively,  and .  If  and  be the - and -intercepts, respectively, of a line, the slope of the line is . This makes the slope of the green line . 

Any line perpendicular to this line must have as its slope the opposite reciprocal of this, or . Since the desired line must also have -intercept , the equation of the line, in point=slope form, is

which can be simplified as

To locate the -intercept of the equation , substitute 0 for :

The -intercept of the parabola is .

The vertex of the parabola of an equation of the form  has -coordinate . Here, we substitute , to obtain -coordinate

.

To find the -coordinate, substitute this for :

The vertex is .

The line includes points  and ; apply the slope formula:

The slope is , and the -intercept is ; in the slope-intercept form , substitute for  and . The equation of the line is .

A line with undefined slope is a vertical line, and its equation is  for some , so the -coordinate of all points it passes through is . If it goes through the vertex of a parabola , then the line has the equation . Therefore, all we need to find is the -coordinate of the vertex of the parabola.

The vertex of the parabola of the equation  has as its -coordinate , which, for the parabola of the equation , can be found by setting :

The desired line is .

 Let  and  be the - and -intercepts, respectively, of the line. Then the slope of the line is , or, equilvalently, . 

We can examine the intercepts in each choice to determine which set meets these conditions.

 and 

Slope:  

 and 

Slope:  

 and 

Slope:  

 and 

Slope:  

 and  comprise the correct choice, since a line passing through these points has the correct slope.

The -intercept of the line coincides with that of the graph of the quadratic equation, which is a parabola; to find the -intercept of the parabola, substitute 0 for  in the quadratic equation:

The -intercept of the parabola, and of the line, is .

The -intercept of the line coincides with one of those of the parabola; to find the -intercepts of the parabola, substitute 0 for  in the equation:

Using the  method, split the middle term by finding two integers whose product is  and whose sum is ; by trial and error we find these to be  and 4, so proceed as follows:

Split:

or

The -intercepts of the parabola are  and , so the -intercept of the line is one of these. We examine both possibilities.

If  and  be the - and -intercepts, respectively, of the line, then the slope of the line is , or, equivalently, 

If the intercepts are  and , the slope is ; if the intercepts are  and , the slope is . Since the line is of negative slope, we choose the line of slope ; since its -intercept is , then we can substitute  in the slope-intercept form of the line, , to get the correct equation, .