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The equation $y = -\frac{1}{2}x + \frac{1}{4}$ 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 a straight line. $\newline$ (B)When the input is $2$ , the output is $-1$ . $\newline$ (C)The rate of change is constant. $\newline$ (D) $y$ is a function of $x$ . $\newline$ (E)The y-intercept is $(0,-\frac{1}{2})$ .

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Write equations of cosine functions using properties

Full solution

Q. The equation

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

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 a straight line.

\newline

(B)When the input is

2

, the output is

-1

.

\newline

(C)The rate of change is constant.

\newline

(D)

y

is a function of

x

.

\newline

(E)The y-intercept is

(0,-\frac{1}{2})

.

Identify Equation Type: First, identify the type of equation. $y = -\frac{1}{2}x + \frac{1}{4}$ is a linear equation because it is in the form $y = mx + b$ , where $m$ and $b$ are constants.
Check Linearity: Check if the graph is a straight line. Since it's a linear equation, its graph will indeed be a straight line.
Calculate Output: Calculate the output when the input $x$ is $2$ . Substitute $x = 2$ into the equation: $\newline$ $y = -\frac{1}{2}(2) + \frac{1}{4} = -1 + \frac{1}{4} = -\frac{3}{4}$ .
Constant Rate of Change: Determine if the rate of change is constant. The coefficient of $x$ , $-\frac{1}{2}$ , represents the slope, which is the rate of change in a linear equation. This rate is constant.
Verify Functionality: Verify if $y$ is a function of $x$ . In the equation $y = -\frac{1}{2}x + \frac{1}{4}$ , for each value of $x$ , there is exactly one corresponding value of $y$ , which satisfies the definition of a function.
Identify Y-Intercept: Identify the y-intercept. The y-intercept is the value of $y$ when $x = 0$ . Substitute $x = 0$ : $y = -\frac{1}{2}(0) + \frac{1}{4} = \frac{1}{4}.$ This shows the y-intercept is $(0, \frac{1}{4})$ , not $(0, -\frac{1}{2})$ .

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