Question
a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
b.Make a table of values that includes the vertex.
c.Use this information to graph the function.
1.f(x)=x^2+4x-7
2.f(x)=-2x^2+5x
3.f(x)=-x^2-6x-9
b.Make a table of values that includes the vertex.
c.Use this information to graph the function.
1.f(x)=x^2+4x-7
2.f(x)=-2x^2+5x
3.f(x)=-x^2-6x-9
Answers
Answered by
GPT-4o mini
Let's analyze each function step-by-step.
### 1. \( f(x) = x^2 + 4x - 7 \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = 0^2 + 4(0) - 7 = -7
\]
So the y-intercept is \((0, -7)\).
- **Axis of Symmetry**: The axis of symmetry for a quadratic function in the form \( ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a} = -\frac{4}{2(1)} = -2
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = -2
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f(-2) \) for the vertex:
\[
f(-2) = (-2)^2 + 4(-2) - 7 = 4 - 8 - 7 = -11
\]
Now, we can create a table with several values:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & (-4)^2 + 4(-4) - 7 = 16 - 16 - 7 = -7 \\
-3 & (-3)^2 + 4(-3) - 7 = 9 - 12 - 7 = -10 \\
-2 & (-2)^2 + 4(-2) - 7 = -11 \\
-1 & (-1)^2 + 4(-1) - 7 = 1 - 4 - 7 = -10 \\
0 & 0^2 + 4(0) - 7 = -7 \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (-4, -7)
- (-3, -10)
- (-2, -11) (vertex)
- (-1, -10)
- (0, -7)
### 2. \( f(x) = -2x^2 + 5x \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = -2(0)^2 + 5(0) = 0
\]
So the y-intercept is \((0, 0)\).
- **Axis of Symmetry**:
\[
x = -\frac{5}{2(-2)} = \frac{5}{4}
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = \frac{5}{4}
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f\left(\frac{5}{4}\right) \):
\[
f\left(\frac{5}{4}\right) = -2\left(\frac{5}{4}\right)^2 + 5\left(\frac{5}{4}\right) = -2\left(\frac{25}{16}\right) + \frac{25}{4} = -\frac{50}{16} + \frac{100}{16} = \frac{50}{16} = \frac{25}{8}
\]
We can create a table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 0 \\
1 & -2(1)^2 + 5(1) = 3 \\
2 & -2(2)^2 + 5(2) = 2 \\
\frac{5}{4} & \frac{25}{8} \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (0, 0)
- (1, 3)
- (2, 2)
- \(\left(\frac{5}{4}, \frac{25}{8}\right)\)
### 3. \( f(x) = -x^2 - 6x - 9 \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = -0^2 - 6(0) - 9 = -9
\]
So the y-intercept is \((0, -9)\).
- **Axis of Symmetry**:
\[
x = -\frac{-6}{2(-1)} = \frac{6}{-2} = -3
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = -3
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f(-3) \):
\[
f(-3) = -(-3)^2 - 6(-3) - 9 = -9 + 18 - 9 = 0
\]
Now create a table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & -(-4)^2 - 6(-4) - 9 = -16 + 24 - 9 = -1 \\
-3 & 0 \\
-2 & -(-2)^2 - 6(-2) - 9 = -4 + 12 - 9 = -1 \\
-1 & -(-1)^2 - 6(-1) - 9 = -1 + 6 - 9 = -4 \\
0 & -9 \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (-4, -1)
- (-3, 0) (vertex)
- (-2, -1)
- (-1, -4)
- (0, -9)
This set of analyses provides you with a comprehensive understanding of the given quadratic functions with their respective graphs and characteristics.
### 1. \( f(x) = x^2 + 4x - 7 \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = 0^2 + 4(0) - 7 = -7
\]
So the y-intercept is \((0, -7)\).
- **Axis of Symmetry**: The axis of symmetry for a quadratic function in the form \( ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a} = -\frac{4}{2(1)} = -2
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = -2
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f(-2) \) for the vertex:
\[
f(-2) = (-2)^2 + 4(-2) - 7 = 4 - 8 - 7 = -11
\]
Now, we can create a table with several values:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & (-4)^2 + 4(-4) - 7 = 16 - 16 - 7 = -7 \\
-3 & (-3)^2 + 4(-3) - 7 = 9 - 12 - 7 = -10 \\
-2 & (-2)^2 + 4(-2) - 7 = -11 \\
-1 & (-1)^2 + 4(-1) - 7 = 1 - 4 - 7 = -10 \\
0 & 0^2 + 4(0) - 7 = -7 \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (-4, -7)
- (-3, -10)
- (-2, -11) (vertex)
- (-1, -10)
- (0, -7)
### 2. \( f(x) = -2x^2 + 5x \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = -2(0)^2 + 5(0) = 0
\]
So the y-intercept is \((0, 0)\).
- **Axis of Symmetry**:
\[
x = -\frac{5}{2(-2)} = \frac{5}{4}
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = \frac{5}{4}
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f\left(\frac{5}{4}\right) \):
\[
f\left(\frac{5}{4}\right) = -2\left(\frac{5}{4}\right)^2 + 5\left(\frac{5}{4}\right) = -2\left(\frac{25}{16}\right) + \frac{25}{4} = -\frac{50}{16} + \frac{100}{16} = \frac{50}{16} = \frac{25}{8}
\]
We can create a table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 0 \\
1 & -2(1)^2 + 5(1) = 3 \\
2 & -2(2)^2 + 5(2) = 2 \\
\frac{5}{4} & \frac{25}{8} \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (0, 0)
- (1, 3)
- (2, 2)
- \(\left(\frac{5}{4}, \frac{25}{8}\right)\)
### 3. \( f(x) = -x^2 - 6x - 9 \)
#### a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
- **y-intercept**: Set \( x = 0 \):
\[
f(0) = -0^2 - 6(0) - 9 = -9
\]
So the y-intercept is \((0, -9)\).
- **Axis of Symmetry**:
\[
x = -\frac{-6}{2(-1)} = \frac{6}{-2} = -3
\]
- **x-coordinate of the vertex**: This is the same as the axis of symmetry:
\[
x = -3
\]
#### b. Make a table of values that includes the vertex.
Calculate \( f(-3) \):
\[
f(-3) = -(-3)^2 - 6(-3) - 9 = -9 + 18 - 9 = 0
\]
Now create a table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & -(-4)^2 - 6(-4) - 9 = -16 + 24 - 9 = -1 \\
-3 & 0 \\
-2 & -(-2)^2 - 6(-2) - 9 = -4 + 12 - 9 = -1 \\
-1 & -(-1)^2 - 6(-1) - 9 = -1 + 6 - 9 = -4 \\
0 & -9 \\
\hline
\end{array}
\]
#### c. Graph the function.
Plot the points from the table:
- (-4, -1)
- (-3, 0) (vertex)
- (-2, -1)
- (-1, -4)
- (0, -9)
This set of analyses provides you with a comprehensive understanding of the given quadratic functions with their respective graphs and characteristics.