### All SAT II Math II Resources

## Example Questions

### Example Question #72 : Functions And Graphs

Note: Figure NOT drawn to scale.

Refer to the above figure. The circle has its center at the origin; the line is tangent to the circle at the point indicated. What is the equation of the line in slope-intercept form?

**Possible Answers:**

Insufficient information is given to determine the equation of the line.

**Correct answer:**

A line tangent to a circle at a given point is perpendicular to the radius from the center to that point. That radius, which has endpoints , has slope

.

The line, being perpendicular to this radius, will have slope equal to the opposite of the reciprocal of that of the radius. This slope will be . Since it includes point , we can use the point-slope form of the line to find its equation:

### Example Question #1 : Circles

What is the center and radius of the circle indicated by the equation?

**Possible Answers:**

**Correct answer:**

A circle is defined by an equation in the format .

The center is indicated by the point and the radius .

In the equation , the center is and the radius is .

### Example Question #1 : Graphing Quadratic Functions And Conic Sections

Give the axis of symmetry of the parabola of the equation

**Possible Answers:**

**Correct answer:**

The line of symmetry of the parabola of the equation

is the vertical line

Substitute :

The line of symmetry is

That is, the line of the equation .

### Example Question #2 : Graphing Quadratic Functions And Conic Sections

Give the -intercept(s) of the parabola of the equation

**Possible Answers:**

The parabola has no -intercept.

and

and

and

**Correct answer:**

and

Set and solve for :

The terms have a GCF of 2, so

The trinomial in parentheses can be FOILed out by noting that and :

Set each of the linear binomials to 0 and solve for :

or

The parabola has as its two intercepts the points and .

### Example Question #3 : Graphing Quadratic Functions And Conic Sections

Give the -coordinate of the vertex of the parabola of the function

**Possible Answers:**

**Correct answer:**

The -coordinate of the vertex of a parabola of the form

is

.

Substitute :

The -coordinate is therefore :

### Example Question #4 : Graphing Quadratic Functions And Conic Sections

A baseball is thrown straight up with an initial speed of 60 miles per hour by a man standing on the roof of a 100-foot high building. The height of the baseball in feet is modeled by the function

To the nearest foot, how high is the baseball when it reaches the highest point of its path?

**Possible Answers:**

**Correct answer:**

We are seeking the value of when the graph of - a parabola - reaches its vertex.

To find this value, we first find the value of . For a parabola of the equation

,

the value of the vertex is

.

Substitute :

The height of the baseball after 1.875 seconds will be

feet.

### Example Question #5 : Graphing Quadratic Functions And Conic Sections

Give the -coordinate of the vertex of the parabola of the function

.

**Possible Answers:**

**Correct answer:**

The -coordinate of the vertex of a parabola of the form

is

.

Set :

The -coordinate is therefore :

, which is the correct choice.

### Example Question #1 : Graphing Functions

A baseball is thrown upward from the top of a one hundred and fifty-foot-high building. The initial speed of the ball is forty-five miles per hour. The height of the ball after seconds is modeled by the function

How high does the ball get (nearest foot)?

**Possible Answers:**

**Correct answer:**

A quadratic function such as has a parabola as its graph. The high point of the parabola - the vertex - is what we are looking for.

The vertex of a function

has as coordinates

.

The second coordinate is the height and we are looking for this quantity. Since , setting :

seconds for the ball to reach the peak.

The height of the ball at this point is , which can be evaluated by substitution:

Round this to 182 feet.

### Example Question #1 : Graphing Trigonometric Functions

Give the amplitude of the graph of the function

**Possible Answers:**

**Correct answer:**

The amplitude of the graph of a sine function is . Here, , so this is the amplitude.

### Example Question #1 : Graphing Trigonometric Functions

Which of these functions has a graph with amplitude 4?

**Possible Answers:**

**Correct answer:**

The functions in each of the choices take the form of a cosine function

.

The graph of a cosine function in this form has amplitude . Therefore, for this function to have amplitude 4, . Of the five choices, only

matches this description.

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