Functions and Graphs

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Math › Functions and Graphs

Questions 1 - 10

What is the vertex of ? Is it a max or min?

$-\frac{5}{6}, \textup{Max}$

$-\frac{5}{6}, \textup{Min}$

$-\frac{5}{3}, \textup{Max}$

$-\frac{5}{3}, \textup{Min}$

$\frac{5}{6}, \textup{Min}$

Explanation

The polynomial is in standard form of a parabola.

To determine the vertex, first write the formula.

$x=-\frac{b}{2a}$

Substitute the coefficients.

$x=-\frac{-5}{2(-3)} = -\frac{5}{6}$

Since the is negative is negative, the parabola opens down, and we will have a maximum.

The answer is: $-\frac{5}{6}, \textup{Max}$

Consider the following two functions:

$f(x) = x^{2}$ and $g(x) = (x+3)^{2} - 6$

How is the function shifted compared with ?

units left, units down

units right, units down

units left, units up

units right, units down

units left, units down

Explanation

The $(x+3)^{2}$ portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

$(x-1)^2+(y-1)^2\geq25$

Given the above circle inequality, which point is not on the edge of the circle?

Explanation

This is a graph of a circle with radius of 5 and a center of (1,1). The center of the circle is not on the edge of the circle, so that is the correct answer. All other points are exactly 5 units away from the circle's center, making them a part of the circle.

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

$(2,0),\ r=6$

$(-2,0),\ r=6$

$(2,0),\ r=36$

$(-2,0),\ r=36$

Explanation

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 .

A baseball is thrown straight up with an initial speed of 50 feet per second by a man standing on the roof of a 120-foot high building. The height of the baseball in feet, as a function of time in seconds , is modeled by the function

$h(t) = -16t^{2}+50t + 120$

To the nearest tenth of a second, how long does it take for the baseball to hit the ground?

$4.7\textrm{ sec}$

$3.1\textrm{ sec}$

$1.6\textrm{ sec}$

$7.5 \textrm{ sec}$

$6.4\textrm{ sec}$

Explanation

When the baseball hits the ground, the height is 0, so we set $h\left ( t\right ) = 0$ . and solve for .

$-16t^{2}+50t + 120 = 0$

This can be done using the quadratic formula:

$t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

Set :

$t = \frac{-50 \pm \sqrt{50^{2}-4(-16)120}}{2(-16)}$

$t = \frac{-50 \pm \sqrt{2,500- (- 7,680)}}{-32}$

$t = \frac{-50 \pm \sqrt{2,500+7,680}}{-32}$

$t = \frac{-50 \pm \sqrt{10,180}}{-32}$

$t \approx \frac{-50 \pm 100.9}{-32}$

One possible solution:

$t \approx \frac{-50 +100.9}{-32} \approx -1.6$

We throw this out, since time must be positive.

The other:

$t \approx \frac{-50 -100.9}{-32} \approx 4.7$

This solution, we keep. The baseball hits the ground in about 4.7 seconds.

Which inequality does this graph represent?
Hyp inequality 1

Explanation

The hyperbola in the question's graph is .

This could be discerned because it subtracts from and not the other way around, putting the zeros on the axis. This narrows the answers down to and .

Test a point to tell which inequality is being used in the graph.

The point is within the shaded region, for example.

, which is greater than 16, so the answer is .

What is the domain and range of the following graph?

Graph for questions

Domain: All real numbers

Range: $y\leq9$

Domain: $y\leq9$

Range: All real numbers

Domain: All real numbers

Range: $x\leq9$

Domain: $x\leq9$

Range: All real numbers

Domain: All real numbers

Range: All real numbers

Explanation

Domain looks at x-values and range looks at y-values.

The x-values appear to continue to go on forever, which suggests the answer:

"all real numbers"

The y-values are all number that are equal to nine or less which is

$y\leq9$

So you answer is:

Domain: All real numbers

Range: $y\leq9$

What are the -intercepts of the equation?

$y=\frac{x^2-9}{x^2-16}$

There are no -intercepts.

Explanation

To find the x-intercepts of the equation, we set the numerator equal to zero.

$\sqrt{9}=\sqrt{x^2}$

What is the center of the circular function ?

Explanation

Remember that the "shifts" involved with circular functions are sort of like those found in parabolas. When you shift a parabola left or right, you have to think "oppositely". A right shift requires you to subtract from the x-component, and a left one requires you to add. Hence, this circle has no horizontal shift, but does shift 6 upward for the vertical component.

You can also remember the general formula for a circle with center at and a radius of .