Question

algebra 2 - 2.1.2 skills practice a
find the characteristics that can be determined from the given equation.

1. $f(x) = 3(x + 2)(x - 4)$
2. $h(x) = -4(x - 4)^2 - 6$
3. $g(x) = 3x^2 - 6x + 9$
4. $k(x) = -4x^2 + 6x$
5. $f(x) = -\frac{1}{2}(x + 3)(x + 12)$
6. $m(x) = 2(x + 8)^2 + 9$
7. $h(x) = \frac{1}{3}(x - 5)^2 - 10$
8. $d(x) = -x^2 - 7x + 11$
9. $g(x) = 11(x - 10)(x - \frac{1}{2})$

find all the characteristics of the given equations.

1. $f(x) = 2(x + 3)^2 - 8$

Explanation:

Let's solve problem 10: \( f(x) = 2(x + 3)^2 - 8 \)

Step 1: Identify the form of the quadratic function

The given function is in vertex form: \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width of the parabola.

Step 2: Find the vertex

In the equation \( f(x) = 2(x + 3)^2 - 8 \), we can rewrite \( (x + 3) \) as \( (x - (-3)) \). So, comparing with \( a(x - h)^2 + k \), we have \( h = -3 \) and \( k = -8 \). Thus, the vertex is \((-3, -8)\).

Step 3: Determine the direction of the parabola

The coefficient \( a = 2 \). Since \( a > 0 \), the parabola opens upward.

Step 4: Find the axis of symmetry

The axis of symmetry for a parabola in vertex form is the vertical line \( x = h \). So, the axis of symmetry is \( x = -3 \).

Step 5: Find the minimum value (since it opens upward)

The minimum value of the function occurs at the vertex. So, the minimum value is \( f(-3) = -8 \).

Step 6: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( f(0) \):
\[

$$\begin{align*} f(0) &= 2(0 + 3)^2 - 8 \\ &= 2(9) - 8 \\ &= 18 - 8 \\ &= 10 \end{align*}$$

\]
So, the y-intercept is \( (0, 10) \).

Step 7: Find the x-intercepts (if any)

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[

$$\begin{align*} 2(x + 3)^2 - 8 &= 0 \\ 2(x + 3)^2 &= 8 \\ (x + 3)^2 &= 4 \\ x + 3 &= \pm 2 \\ x &= -3 \pm 2 \end{align*}$$

\]
So, \( x = -3 + 2 = -1 \) or \( x = -3 - 2 = -5 \). Thus, the x-intercepts are \( (-5, 0) \) and \( (-1, 0) \).

Final Answer for Problem 10:

• Vertex: \((-3, -8)\)
• Axis of Symmetry: \( x = -3 \)
• Direction of Opening: Upward (since \( a = 2 > 0 \))
• Minimum Value: \(-8\) (occurring at \( x = -3 \))
• Y-intercept: \((0, 10)\)
• X-intercepts: \((-5, 0)\) and \((-1, 0)\)

Answer:

Let's solve problem 10: \( f(x) = 2(x + 3)^2 - 8 \)

Step 1: Identify the form of the quadratic function

The given function is in vertex form: \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width of the parabola.

Step 2: Find the vertex

In the equation \( f(x) = 2(x + 3)^2 - 8 \), we can rewrite \( (x + 3) \) as \( (x - (-3)) \). So, comparing with \( a(x - h)^2 + k \), we have \( h = -3 \) and \( k = -8 \). Thus, the vertex is \((-3, -8)\).

Step 3: Determine the direction of the parabola

The coefficient \( a = 2 \). Since \( a > 0 \), the parabola opens upward.

Step 4: Find the axis of symmetry

The axis of symmetry for a parabola in vertex form is the vertical line \( x = h \). So, the axis of symmetry is \( x = -3 \).

Step 5: Find the minimum value (since it opens upward)

The minimum value of the function occurs at the vertex. So, the minimum value is \( f(-3) = -8 \).

Step 6: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( f(0) \):
\[

$$\begin{align*} f(0) &= 2(0 + 3)^2 - 8 \\ &= 2(9) - 8 \\ &= 18 - 8 \\ &= 10 \end{align*}$$

\]
So, the y-intercept is \( (0, 10) \).

Step 7: Find the x-intercepts (if any)

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[

$$\begin{align*} 2(x + 3)^2 - 8 &= 0 \\ 2(x + 3)^2 &= 8 \\ (x + 3)^2 &= 4 \\ x + 3 &= \pm 2 \\ x &= -3 \pm 2 \end{align*}$$

\]
So, \( x = -3 + 2 = -1 \) or \( x = -3 - 2 = -5 \). Thus, the x-intercepts are \( (-5, 0) \) and \( (-1, 0) \).

Final Answer for Problem 10:

• Vertex: \((-3, -8)\)
• Axis of Symmetry: \( x = -3 \)
• Direction of Opening: Upward (since \( a = 2 > 0 \))
• Minimum Value: \(-8\) (occurring at \( x = -3 \))
• Y-intercept: \((0, 10)\)
• X-intercepts: \((-5, 0)\) and \((-1, 0)\)