Here is a breakdown of the math problems listed in the Homework:

Problem 1

If $(t, -2)$ lies on the graph of $y = x^2 - 3x - 2$, find $t$.

• Substitute $y = -2$ into the equation $y = x^2 - 3x - 2$:
  $-2 = t^2 - 3t - 2$ Solve for $t$.

Problem 2

Using a table of values from $x = -3$ to $x = 3$, draw the graph of $y = x^2 - 2x + 3$.

• Create a table of values by substituting $x = -3, -2, \dots, 3$ into the equation:
  $y = x^2 - 2x + 3$
  Plot the points on a graph.

Problem 3

Using a translation, sketch the graph of $y = -(x + 1)^2 + 3$.

• Recognize that this is a parabola translated 1 unit left and 3 units up.
• The vertex is $(-1, 3)$.

Problem 4

(a) Write $y = x^2 + 2x + 1$ in vertex form $y = (x - h)^2 + k$ by completing the square.

• Rearrange the expression:
  $y = x^2 + 2x + 1 = (x + 1)^2$
  So, the vertex form is $y = (x + 1)^2$.

(b) State the vertex coordinates.

• The vertex is $(-1, 0)$.

(c) Sketch the graphs of $y = x^2$ and $y = x^2 + 2x + 1$ on the same set of axes.

Problem 5

(a) Write $y = -x^2 + 4x$ in vertex form $y = -(x - h)^2 + k$ by completing the square.

• Factor out $-1$:
  $y = -\left(x^2 - 4x\right) = -\left((x - 2)^2 - 4\right)$
  Simplify:
  $y = -(x - 2)^2 + 4$

(b) State the vertex coordinates.

• The vertex is $(2, 4)$.

(c) Sketch the graphs of $y = -x^2$ and $y = -x^2 + 4x$ on the same set of axes.

Problem 6

Find the $x$-intercepts of:

(a) $y = -(x + 2)(x - 5)$:

• Set $y = 0$:
  $0 = -(x + 2)(x - 5)$
  Solve for $x$:
  $x = -2 \quad \text{or} \quad x = 5$

(b) $y = (x + 2)^2$:

• Set $y = 0$:
  $0 = (x + 2)^2$
  Solve for $x$:
  $x = -2$

Would you like me to go into detail for any of these steps? Let me know!

Related Questions

1. How do you complete the square for a quadratic expression?
2. How do you find the vertex of a parabola given its equation?
3. How do you graph a parabola from its vertex form?
4. How can you determine the x-intercepts of a quadratic equation?
5. What are the effects of $a$, $h$, and $k$ in the vertex form $y = a(x - h)^2 + k$?

Tip: Remember that completing the square helps convert standard form equations into vertex form, making it easier to analyze the graph.