Explanation:

Letandbe 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,andare of the same sign. Since the quotient of two numbers of the same sign is positive, it follows thatis negative, and therefore, none of the pairs of intercepts can be those of a line with positive slope.

Explanation:

Letandbe 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:

andcomprise the correct choice.

Explanation:

The slope of the green line can be calculated by noting that the- and-intercepts of the line are, respectively,and. Ifandbe 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

Explanation:

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 formhas-coordinate. Here, we substitute, to obtain-coordinate

.

To find the-coordinate, substitute this for:

The vertex is.

线包括分and; apply the slope formula:

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

Explanation:

A line with undefined slope is a vertical line, and its equation isfor 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 equationhas as its-coordinate, which, for the parabola of the equation, can be found by setting:

The desired line is.

Explanation:

Letandbe 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:

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

Explanation:

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 forin 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 forin the equation:

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

Split:

or

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

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

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

Explanation:

Ifandbe the- and-intercepts, respectively, of a line, the slope of the line is.

The- and-intercepts of the line are, respectively,and, so, and consequently, the slope of the green line is. A line parallel to this line must also have slope.

Each of the equations of the lines is in slope-intercept form, whereis the slope, so we need only look at the coefficients of. The only choice that hasas its-coefficient is, so this is the correct choice.

Explanation:

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 forin 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 forin the equation:

The quadratic expression can be "reverse-FOILed" by noting that 9 andhave productand sum 7:

, in which case

or

, in which case.

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

Ifandbe the- and-intercepts, respectively, of the line, then the slope of the line is. If the intercepts areand, the slope is; if the intercepts areand, the slope is. Since the line is of positive slope, we choose the line of slope 9; since its-intercept is, then we can substitutein the slope-intercept form of the line,, to get the correct equation,.

Explanation:

We can find out whether the graphs ofandintersect by first solving forin the first equation:

We then substitute in the second equation for:

Then we rewrite in standard form:

Since we are only trying to pdetermine whether at least one point of intersection exists, rather than actually find the point, all we need to do is to evaluate the discriminant; if it is nonnegative, at least one solution - and, consequently, one point of intersection - exists. In the general quadratic equation, this is, so here, the discriminant is

.

Therefore, the line of the equationintersects the parabola of the equation.

我们做the same for the other three lines:

Then we rewrite in standard form:

.

The line ofintersects the parabola.

The line ofintersects the parabola.

Since the discriminant is negative, the system has no real solution. This means that the line ofdoes not intersect the parabola of the equation, and it is the correct choice.