Parabolic Functions

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Algebra 2 › Parabolic Functions

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Consider the equation:

$y = 3x^{2} - 10x +5$

The vertex of this parabolic function would be located at:

$(\frac{5}{3}, \frac{-10}{3})$

CORRECT

$(\frac{-5}{3}, \frac{-10}{3})$

$(\frac{-5}{3}, 30)$

$(\frac{5}{3}, \frac{2}{25})$

$(\frac{5}{3}, \frac{27}{25})$

Explanation

For any parabola, the general equation is

$y = ax^{2} + bx +c$ , and the x-coordinate of its vertex is given by

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

For the given problem, the x-coordinate is

$x = \frac{-(-10)}{2(3)} = \frac{5}{3}$ .

To find the y-coordinate, plug $\frac{5}{3}$ into the original equation:

$y = 3(\frac{5}{3})^{2} - 10(\frac{5}{3}) +5 = \frac{-10}{3}$

Therefore the vertex is at $(\frac{5}{3}, \frac{-10}{3})$ .

What are the -intercepts of the equation?

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

CORRECT

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 are the -intercepts of the equation?

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

CORRECT

There are no -intercepts.

Explanation

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

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

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

CORRECT

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 .

Based on the figure below, which line depicts a quadratic function?

Red line

CORRECT

Blue line

Green line

Purple line

None of them

Explanation

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to $\small {f(x)=ax^2+bx+c}$ .

The blue line represents a linear function and will have a formula similar to $\small {f(x)=ax+b$ .

The green line represents an exponential function and will have a formula similar to $\small {f(x)=ax^b}$ .

The purple line represents an absolute value function and will have a formula similar to $\small {f(x)=\left |ax+b \right |}.$ .

Consider the equation:

$y = 3x^{2} - 10x +5$

The vertex of this parabolic function would be located at:

$(\frac{5}{3}, \frac{-10}{3})$

CORRECT

$(\frac{-5}{3}, \frac{-10}{3})$

$(\frac{-5}{3}, 30)$

$(\frac{5}{3}, \frac{2}{25})$

$(\frac{5}{3}, \frac{27}{25})$

Explanation

For any parabola, the general equation is

$y = ax^{2} + bx +c$ , and the x-coordinate of its vertex is given by

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

For the given problem, the x-coordinate is

$x = \frac{-(-10)}{2(3)} = \frac{5}{3}$ .

To find the y-coordinate, plug $\frac{5}{3}$ into the original equation:

$y = 3(\frac{5}{3})^{2} - 10(\frac{5}{3}) +5 = \frac{-10}{3}$

Therefore the vertex is at $(\frac{5}{3}, \frac{-10}{3})$ .

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

CORRECT

units right, units down

units left, units up

units right, units down

units left, units down

Explanation

Based on the figure below, which line depicts a quadratic function?

Red line

CORRECT

Blue line

Green line

Purple line

None of them

Explanation

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to $\small {f(x)=ax^2+bx+c}$ .

The blue line represents a linear function and will have a formula similar to $\small {f(x)=ax+b$ .

The green line represents an exponential function and will have a formula similar to $\small {f(x)=ax^b}$ .

The purple line represents an absolute value function and will have a formula similar to $\small {f(x)=\left |ax+b \right |}.$ .

What is the equation of a parabola with vertex and -intercept ?

$y = -\frac{1}{2} x^{2} +4x - 7$

CORRECT

$y = -\frac{1}{2} x^{2} -4x - 7$

$y = \frac{1}{2} x^{2} -4x - 7$

$y = \frac{1}{2} x^{2} -8x - 7$

$y = \frac{1}{2} x^{2} +8x - 7$

Explanation

From the vertex, we know that the equation of the parabola will take the form $y = a (x - 4) ^{2}+ 1$ for some .

To calculate that , we plug in the values from the other point we are given, , and solve for :

$y = a (x - 4)^{2} + 1$

$-7= a \cdot (0 - 4)^{2} + 1$

$-7= a \cdot (- 4)^{2} + 1$

$a = -\frac{1}{2}$

Now the equation is $y = -\frac{1}{2} (x - 4)^{2} + 1$ . This is not an answer choice, so we need to rewrite it in some way.

Expand the squared term:

$y = -\frac{1}{2} (x^{2} -8x + 16) + 1$

Distribute the fraction through the parentheses:

$y = -\frac{1}{2} x^{2} +4x - 8 + 1$

Combine like terms:

$y = -\frac{1}{2} x^{2} +4x - 7$

Which of the following functions represents a parabola?

$f(x) = x^{2} - 9$

CORRECT

$f(x) = \frac{4x^{2}}{x^{3}}$

$f(x) = x^{2} - y^{2}$

Explanation

A parabola is a curve that can be represented by a quadratic equation. The only quadratic here is represented by the function $f(x) = x^{2} - 9$ , while the others represent straight lines, circles, and other curves.

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