Review Material in Pre Calculus

PRE-CALCULUS

​Review Materials for R.E.F.R.E.S.H. Club

Prepared by
ANABEL S. SOLLANO
SST III, Mathematics

• Conic Sections

• Exponential and Logarithmic Functions

• Trigonometric Identities

Topic Outline

Multiple Choice

If you cut a double-napped cone horizontally using a plane figure, what type of conic section will be formed?

• If the plane is parallel to the base, it creates a circle.

• A plane parallel to the axis will yield a parabola.

• A hyperbola is formed because the cutting plane crosses both nappes of the cone.

• It creates an ellipse because the distance from the center of the ellipse to any point on the cut is constant.

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Multiple Choice

Consider an ellipse with its major axis aligned along the x-axis. Which of the following statements is correct regarding the properties of this ellipse?

• The center of the ellipse is at the origin.

• The foci of the ellipse are located on the x-axis.

• The distance between the center and each focus is equal to the length of the major axis.

• The latera recta are perpendicular to the  x- axis and passes through the center.

https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/

Multiple Choice

What number completes the equation of the circle $$x^2+y^2=_{ }$$ ______, if $$r=3\sqrt[]{2}$$ ?

• $$3\sqrt[]{2}$$

• 6

• 12

• 18

Draw

What is the general form of the equation of the circle $$\left(x-5\right)^2+\left(y+3\right)^2=9?$$

Draw

Convert the equation of the circle $$4x^2+4y^2-24x+32y-4=0$$ in its standard form.

Multiple Choice

What are the center and radius of the circle $$\left(x-5\right)^2+\left(y+3\right)^2=4$$ ?

• C: (-5,3); r = 4

• C: (5, -3); r = 4

• C: (5, -3); r = 2

• C: (-5, 3); r = 2

A street with two lanes, each 10 ft wide, goes through a semicircular tunnel with radius 12 ft. How high is the tunnel at the edge of each lane?

The arch of an underpass is a semi-ellipse which is 10 meters wide and 4 meters high. Write an equation to determine the clearance at the edge of a lane if the edge is 3 meters from the middle.

Multiple Choice

Arrange the following steps involved in solving for the value of the unknown variable in the exponential equation $$2^{5x}=64$$  in the correct order:
i. Equate the exponents on both sides of the equation.

                 ii. Solve for the unknown variable.

                 iii. Express 64 as $$2^6$$ .

• ii, i, iii

• iii, i, ii

• iii, ii, i

• ii, iii, i

Multiple Choice

Consider the exponential equation $$2^x=16$$ . Which of the following logarithmic expressions accurately represents the given equation?

• $$x=\log_2\left(16\right)$$

• $$2=\log_{16}\left(x\right)$$

• $$16=\log_2\left(x\right)$$

• $$\log_{16}\left(2\right)=x$$

Multiple Choice

The price of the iPhone 15 Pro Max is ₱108,990.00. According to Northladder, older models of the iPhone lose an average of 30% of their value during the weeks of a new launch. Assuming that Apple releases new models every year, which equation best describes the value of the iPhone 15 Pro Max in t years?

• $$y=108,990\left(30\right)^t$$

• $$y=108,990\left(0.30\right)^t$$

• $$y=108,990\left(1.30\right)^t$$

• $$y=108,990\left(0.7\right)^t$$

Multiple Choice

In a wildlife reserve, there were initially 75 wolves. Over the course of nine months, the wolf population increased to 225. Assuming exponential growth, what is the projected number of wolves at the end of three years?

• 6,000

• 2,000

• 1,000

• 300

Does the mapping diagram at the right represent a function?

Multiple Choice

How do you know if a function, $$f\left(x\right)$$ , has an inverse?

• It is a one-to-one function, ensuring each input corresponds to a unique output.

• The graph should pass both the horizontal and vertical line tests.

• The composition of the functions should equate to x.

• All of the above.

Which graph represents a one-to-one function?

Draw

What value or values satisfy the equation $$\log_9\left(x^2-8x\right)=1?$$ ?

Multiple Choice

The magnitude 9.5 earthquake that occurred in Valdivia, Chile, on May 22, 1960, holds the record as the largest earthquake in world history. This devastating event resulted in the loss of 1655 lives, injuries to 3000 individuals, and the displacement of two million people. Additionally, it triggered a tsunami that caused fatalities and extensive damage as far away as Hawaii, Japan, and the Philippines. How does the intensity of this earthquake compare to the 7.9 magnitude earthquake in Moro Gulf, Mindanao, which is considered the most severe earthquake in the Philippines?

• The earthquake in Chile is 2 times more intense than the earthquake in Mindanao.

• The earthquake in Chile is 251 times more intense than the earthquake in Mindanao.

• The earthquake in Chile is 7 times more intense than the earthquake in Mindanao.

• The earthquake in Chile is 40 times more intense than the earthquake in Mindanao.

Multiple Choice

Which expression is equivalent to 1?

• $$\frac{1+\sin\theta}{\sin\ \theta}$$

• $$\frac{1}{\sec^2\theta}+\frac{1}{\csc^2\theta}$$

• $$\tan^2\theta-\sec^2\theta$$

• $$\frac{\cot\theta\csc\theta}{\sec\theta}$$

Draw

Simplify the trigonometric expression $$\frac{\cot\theta\sec\theta}{\csc\theta}$$ .

Multiple Choice

If you were asked to verify the identity $$\frac{\sin^2\theta+\cos^2\theta}{\cos\theta}=\sec\theta$$ , what would be the most appropriate initial step to undertake?

• Replace $$\cos\theta$$ on the left side of the equation with $$\frac{1}{\sec\theta}$$

• Replace $$\sin^2\theta+\cos^2\theta$$ on the left side of the equation with 1.

• Replace $$\sin^2\theta$$ on the left side of the equation with $$\frac{1}{\csc^2\theta}$$

• You can't verify the identity because it isn't true.

A student was asked to verify the following identity. Where did the student make a mistake or is the student correct?