Gr 10 Math Ch 5: Hyperbolic Function

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What is the domain of the hyperbolic function defined by the equation $y = \frac{a}{x} + q$?

\{x : x \in \mathbb{R}, x > 0\}

\{x : x \in \mathbb{R}, x \neq 0\} (correct)

\{x : x \in \mathbb{R}\}

\{x : x \in \mathbb{R}, x = 0\}

What happens to the graph of the hyperbolic function when $q > 0$?

The graph is reflected across the x-axis.

The graph is shifted vertically upwards by $q$ units. (correct)

No change occurs to the graph.

The graph is shifted horizontally to the right.

What is the nature of the vertical asymptote for a hyperbolic function?

The line $x = 0$. (correct)

The line $y = 0$.

The line $y = a$.

The line $y = q$.

When $a < 0$ in the hyperbolic function $y = \frac{a}{x} + q$, in which quadrants does the graph lie?

Second and fourth quadrants. Signup and view all the answers

Which of the following statements about the y-intercept of the hyperbolic function is true?

The function does not have a y-intercept. Signup and view all the answers

To find the x-intercept of the function $y = \frac{a}{x} + q$, what equation must be solved?

$y = 0$. Signup and view all the answers

Which line represents the horizontal asymptote of the hyperbolic function?

The line $y = q$. Signup and view all the answers

What effect does changing $a$ from positive to negative have on the hyperbolic function's shape?

It switches the graph from the third and first to the second and fourth quadrants. Signup and view all the answers

What is the range of the hyperbolic function defined by the equation $y = \frac{a}{x} + q$?

{y : y \in \mathbb{R}, y \neq q} Signup and view all the answers

Which quadrants does a hyperbolic function graph occupy if $a$ is positive?

First and third quadrants Signup and view all the answers

What is the effect of a vertical shift when $q < 0$ in the hyperbolic function?

The graph shifts downwards Signup and view all the answers

Which of the following lines serves as the vertical asymptote for the hyperbolic function?

$x = 0$ Signup and view all the answers

How can the x-intercept of the hyperbolic function be determined?

By setting $y$ to zero and solving for $x$ Signup and view all the answers

What characteristics describe a hyperbolic function's y-intercept?

It is undefined. Signup and view all the answers

If $q$ is greater than zero in the hyperbolic function, how does the graph behave?

It is shifted vertically upwards. Signup and view all the answers

Which of the following is a characteristic of the hyperbolic function's graph?

It has symmetry along the lines $y = x + q$ and $y = -x + q$. Signup and view all the answers

What describes the axes of symmetry for the hyperbolic function?

The lines y = x + q and y = -x + q Signup and view all the answers

How is the x-intercept of the hyperbolic function determined?

By setting $y = 0$ and solving for x Signup and view all the answers

What would happen to the graph of a hyperbolic function if $q$ is assigned a negative value?

The graph shifts downwards by |q| units Signup and view all the answers

Which statement is true regarding the horizontal asymptote of a hyperbolic function?

The horizontal asymptote is represented by the line $y = q$ Signup and view all the answers

For the hyperbolic function $y = rac{a}{x} + q$, when is the function considered undefined?

When x = 0 Signup and view all the answers

What is the implication of a positive value for $a$ in a hyperbolic function?

The graph lies in the first and third quadrants Signup and view all the answers

Which statement about the range of the hyperbolic function is accurate?

The range excludes q but includes all other real numbers Signup and view all the answers

In a hyperbolic function, how does the value of q affect the horizontal asymptote?

It determines the vertical position of the horizontal asymptote Signup and view all the answers

What can be said about the x-intercept of the hyperbolic function?

It can be found by calculating where the function equals zero. Signup and view all the answers

Which of the following reflects the implications of a vertical asymptote for the hyperbolic function?

The function approaches infinity as x approaches zero. Signup and view all the answers

Which of these properties would you expect for a hyperbolic function with negative value of q?

The graph will be shifted vertically downwards by |q| units. Signup and view all the answers

What is the significance of the axes of symmetry for the hyperbolic function?

They help in understanding the overall shape of the graph. Signup and view all the answers

How does the sign of the coefficient a in the hyperbolic function affect the graph?

It determines the quadrant locations for the graph. Signup and view all the answers

For the hyperbolic function, how is the behavior of the function as x approaches a value of zero characterized?

The function trends toward infinite positive or negative values. Signup and view all the answers

What statement accurately describes the relationship between a and q in the hyperbolic function?

They can be independent of each other's effects on the graph. Signup and view all the answers

If the graph of the hyperbolic function does not have an x-intercept, what can be inferred?

The Graph remains above or below the x-axis without crossing it. Signup and view all the answers