find the decreasing functions and arrange them in order from least to greatest $y$-intercept.

$y = 2.5x + 10$
$y = -0.5x + 0.5$
$y = -4.5x + 3$
$y = -0.5x - 5$
$y = -2x + 7$
$y = -5x - 12$

Question

find the decreasing functions and arrange them in order from least to greatest $y$-intercept.

$y = 2.5x + 10$
$y = -0.5x + 0.5$
$y = -4.5x + 3$
$y = -0.5x - 5$
$y = -2x + 7$
$y = -5x - 12$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Slope of a Line", "Linear Equations" ], "new_concepts": [], "current_concepts": [ "Slope of a Line", "Linear Equations", "Linear Functions" ] } ## Identify decreasing functions $$ \begin{aligned} &\text{A linear function } y = mx + b \text{ is decreasing if } m < 0.\ &y = 2.5x + 10 \implies m = 2.5 > 0 \quad (\text{increasing})\ &y = -0.5x + 0.5 \implies m = -0.5 < 0 \quad (\text{decreasing})\ &y = -4.5x + 3 \implies m = -4.5 < 0 \quad (\text{decreasing})\ &y = -0.5x - 5 \implies m = -0.5 < 0 \quad (\text{decreasing})\ &y = -2x + 7 \implies m = -2 < 0 \quad (\text{decreasing})\ &y = -5x - 12 \implies m = -5 < 0 \quad (\text{decreasing}) \end{aligned} $$ ## Determine y-intercepts of decreasing functions $$ \begin{aligned} &y = -5x - 12 \implies b = -12\ &y = -0.5x - 5 \implies b = -5\ &y = -0.5x + 0.5 \implies b = 0.5\ &y = -4.5x + 3 \implies b = 3\ &y = -2x + 7 \implies b = 7 \end{aligned} $$ ## Order from least to greatest y-intercept $$ \begin{aligned} &-12 < -5 < 0.5 < 3 < 7\ &\text{Ordered functions:}\ &1. \quad y = -5x - 12\ &2. \quad y = -0.5x - 5\ &3. \quad y = -0.5x + 0.5\ &4. \quad y = -4.5x + 3\ &5. \quad y = -2x + 7 \end{aligned} $$ The decreasing functions ordered from least to greatest $y$-intercept are: 1. $y = -5x - 12$ 2. $y = -0.5x - 5$ 3. $y = -0.5x + 0.5$ 4. $y = -4.5x + 3$ 5. $y = -2x + 7$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Linear Functions" ] }

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Question

Explanation:

Identify decreasing functions

Determine y-intercepts of decreasing functions

Order from least to greatest y-intercept

Answer: