Let me analyze the content of the uploaded image and extract the mathematical problems labeled as 21, 22, 23, and 24. I’ll provide detailed solutions for these problems. Please hold on for a moment.The problems labeled as 21, 22, 23, and 24 from the uploaded image appear to involve functions, graphs, and equations. Here’s a breakdown and solution for each problem based on their visible structure:

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21. Problem:

Write the equations of the graphs for the functions:

1. $y = (x - 5)^2 - 6$
2. $y = -2x^2 + 3x - 2$

Solution for 21:

1. $y = (x - 5)^2 - 6$:

   • The function represents a parabola shifted:
     • 5 units to the right (due to $x - 5$),
     • 6 units downward (due to $-6$).
   • Vertex: $(5, -6)$,
   • The parabola opens upwards (since the coefficient of $x^2$ is positive).

2. $y = -2x^2 + 3x - 2$:

   • This is a parabola that:
     • Opens downwards (since the coefficient of $x^2$ is negative),
     • Its vertex can be found using $x = -\frac{b}{2a}$ formula:
       • Here, $a = -2$, $b = 3$,
       • $x = -\frac{3}{2(-2)} = \frac{3}{4}$.
       • Plugging $x = \frac{3}{4}$ back into the equation gives $y$.
   • Use standard vertex and axis symmetry analysis for graphing.

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22. Problem:

Graphically represent the following functions:

1. $y = x^2 - 4x + 4$ (simplifies to $y = (x - 2)^2$),
2. $y = 2(x - 1)^2 + 2$.

Solution for 22:

1. $y = x^2 - 4x + 4$:

   • Simplifies to $y = (x - 2)^2$, which is a parabola:
     • Vertex: $(2, 0)$,
     • Opens upwards (since the coefficient of $x^2$ is positive).

2. $y = 2(x - 1)^2 + 2$:

   • This is a parabola:
     • Vertex: $(1, 2)$,
     • Opens upwards with a steeper slope (due to the factor of 2).

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23. Problem:

For the function $y = -x^2 + 4x - 3$:

• Find its graph and analyze its symmetry.

Solution for 23:

• This is a downward-opening parabola ($-x^2$).
• Use $x = -\frac{b}{2a}$ to find the vertex:
  • $a = -1$, $b = 4$,
  • $x = -\frac{4}{2(-1)} = 2$.
• Vertex: $(2, y)$, where $y = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$,
  • Vertex: $(2, 1)$.
• Symmetry: The graph is symmetric about the vertical line $x = 2$.

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24. Problem:

Analyze $y = (x + 2)^2 - 1$:

• Identify its vertex, axis of symmetry, and graph.

Solution for 24:

• The function $y = (x + 2)^2 - 1$ is a parabola:
  • Vertex: $(-2, -1)$ (shifted 2 units left and 1 unit down),
  • Opens upwards (since the coefficient of $x^2$ is positive),
  • Axis of symmetry: $x = -2$.

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If you'd like, I can plot these functions or provide further explanations for any part. Let me know!

Here are five related questions for deeper understanding:

1. How do shifts affect the graph of a quadratic function?
2. What is the significance of the vertex in graphing parabolas?
3. How can you determine the direction a parabola opens from its equation?
4. What role does the axis of symmetry play in analyzing quadratic functions?
5. Can you use the discriminant ($b^2 - 4ac$) to deduce anything about these graphs?

Tip: Always rewrite quadratic equations in vertex form to easily identify their properties.