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The equation $y = 2x^4 - 1$ defines a relationship between $x$ and $y$ , where $x$ is the input and $y$ is the output. Which statements about the relationship are true? Select all that apply. $\newline$ Multi-select Choices: $\newline$ (A)The graph is curved. $\newline$ (B)The y-intercept is $(-1,0)$ . $\newline$ (C)When the input is $2$ , the output is $31$ . $\newline$ (D)The function is linear. $\newline$ (E)The rate of change is not constant.

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One-step inequalities: word problems

Full solution

Q. The equation

y = 2x^4 - 1

defines a relationship between

x

and

y

, where

x

is the input and

y

is the output. Which statements about the relationship are true? Select all that apply.

\newline

Multi-select Choices:

\newline

(A)The graph is curved.

\newline

(B)The y-intercept is

(-1,0)

.

\newline

(C)When the input is

2

, the output is

31

.

\newline

(D)The function is linear.

\newline

(E)The rate of change is not constant.

Determine Graph Shape: Analyze the equation $y = 2x^4 - 1$ to determine the graph's shape. Since the highest power of $x$ is $4$ , which is even, the graph is symmetric and curved like a parabola but steeper.
Identify Y-Intercept: Identify the y-intercept by substituting $x = 0$ into the equation. $y = 2(0)^4 - 1 = -1$ . The y-intercept is $(0, -1)$ , not $(-1, 0)$ .
Calculate Output for $x=2$ : Calculate the output when the input $x$ is $2$ . $y = 2(2)^4 - 1 = 2(16) - 1 = 32 - 1 = 31$ . This confirms that when $x$ is $2$ , $y$ is indeed $31$ .
Check Linearity: Determine if the function is linear by checking if the equation is of the form $y = mx + b$ . Since the equation is $y = 2x^4 - 1$ , it is not linear because it involves $x$ to the fourth power.
Evaluate Rate of Change: Evaluate if the rate of change is constant by considering the derivative. The derivative of $y = 2x^4 - 1$ is $\frac{dy}{dx} = 8x^3$ , which depends on the value of $x$ , indicating that the rate of change is not constant.

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