Functions and Graphs - SAT Subject Test in Math I

Let's start by looking at the given equation:

The inequality is written in slope-intercept form; therefore, the slope is equal to and the y-intercept is equal to .

All of the graphs depict a line with slope of and y-intercept . Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin . If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.

Because this statement is true, the origin must be included in the shaded region, so we shade below the line.

Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.

A horizontal line has equation for some value of ; since the line goes through a point with -coordinate 3, the line is . Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is .

A vertical line has equation for some value of ; since the line goes through a point with -coordinate 4, the line is . Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is .

Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is .

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain the equation of the boundary line:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - for ease, we will choose :

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0 is less than 3 so the correct symbol is .

The inequality is .

To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.

Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.

Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:

which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:

One way to answer this is to first find the equation of the line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The line has slope 3 and -intercept , so we can substitute in the slope-intercept form:

Now substitute 4 for and for and solve for :

First, we need to find the slope of the above line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute and in the slope-intercept form:

First, we need to find the slope of the line.

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown .

We can use this to find the -intercept using the slope formula as follows:

The lower left point has coordinates . Therefore, we can set up and solve for in this slope formula, setting :

We calculate the slopes of the lines using the slope formula.

The slope of line is

The slope of line is

The lines have the same slope, making them either parallel or identical.

Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form , where is the -coordinate of each point on the line. Therefore, line and line have equations and .This makes them parallel lines.

In form, where y = maximum heart rate and x = age, we can express the relationship as:

We are looking for a graph with a slope of -1 and a y-intercept of 220.

The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.

Lines are perpendicular when their slopes are the negative recicprocals of each other such as . To find the slope of our equation we must change it to slope y-intercept form.

Subtract the x variable from both sides:

Divide by 4 to isolate y:

The negative reciprocal of the above slope: . The only equation with this slope is .

We have the following answer choices.

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.

The highest exponent of the variable term is two (). This tells that this function is quadratic, meaning that it is a parabola.

The graph below will be the answer, as it shows a parabolic curve.

crosses the axis when equals 0. So, substitute in 0 for :

The zeros of the parabola are at and , so when placed into the formula

,

each of their signs is reversed to end up with the correct sign in the answer. The coefficient can be found by plugging in any easily-identifiable, non-zero point to the above formula. For example, we can plug in which gives

To simplify, we must first simplify the absolute values.

Now, combine like terms:

The way to figure out this problem is by understanding behavior of polynomials.

The sign that occurs before the is positive and therefore it is understood that the function will open upwards. the "8" on the function is an even number which means that the function is going to be u-shaped. The only answer choice that fits both these criteria is:

Define . Then, if , it follows that .

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values:

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some .

Start by visualizing the graph associated with the function :

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of looks like this:

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function :

Factor 2 as GCF from the first two terms giving us:

Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because ) resulting in the following equation:

which is equal to

Hence the vertex is located at