To graph the functions $h(x) = |x+5|$ and $t(x) = |2x+5|$, let’s analyze their properties step by step.

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1. Graph $h(x) = |x+5|$:

• Vertex: The expression $x+5$ equals zero when $x = -5$. This is the point where the absolute value "turns."
• Behavior:
  • For $x > -5$: $h(x) = x+5$ (linear with positive slope $+1$).
  • For $x < -5$: $h(x) = -(x+5) = -x-5$ (linear with negative slope $-1$).
• Key Points:
  • Vertex: $(-5, 0)$
  • Other points:
    • $h(-6) = 1$, $h(-4) = 1$ (symmetric about $x = -5$).
    • $h(-7) = 2$, $h(-3) = 2$.

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2. Graph $t(x) = |2x+5|$:

• Vertex: The expression $2x+5$ equals zero when $x = -\frac{5}{2} = -2.5$. This is the point where the absolute value "turns."
• Behavior:
  • For $x > -2.5$: $t(x) = 2x+5$ (linear with slope $+2$).
  • For $x < -2.5$: $t(x) = -(2x+5) = -2x-5$ (linear with slope $-2$).
• Key Points:
  • Vertex: $(-2.5, 0)$
  • Other points:
    • $t(-3) = 1$, $t(-2) = 1$ (symmetric about $x = -2.5$).
    • $t(-4) = 3$, $t(-1) = 3$.

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3. Graphing Both Functions Together:

• $h(x) = |x+5|$ has a V-shape centered at $(-5, 0)$.
• $t(x) = |2x+5|$ has a narrower V-shape (due to the coefficient $2$) centered at $(-2.5, 0)$.
• Both graphs open upward and are symmetric about their respective vertices.

Would you like a visual graph for these functions? If so, let me know! 😊

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Related Questions:

1. How do the slopes of the linear sections of $h(x)$ and $t(x)$ differ?
2. What happens to the graphs if we shift them vertically by a constant $c$?
3. How would the graphs change if we replaced $x$ with $-x$ in both functions?
4. What is the domain and range of $h(x)$ and $t(x)$?
5. Can you identify points where $h(x) = t(x)$?

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Tip:

When graphing absolute value functions, always start by identifying the vertex and then analyze how the slopes behave to the left and right.