Question

examples 1-2 complete parts a–c for each quadratic function.
a. find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the
b. make a table of values that includes the vertex.
c. use this information to graph the function.

1. $f(x) = 4x^2$ 13. $f(x) = -2x^2$
2. $f(x) = x^2 - 5$ 15. $f(x) = x^2 + 3$
3. $f(x) = 4x^2 - 3$ 17. $f(x) = -3x^2 + 5$
4. $f(x) = x^2 - 6x + 8$ 19 $f(x) = x^2 - 3x - 10$
5. $f(x) = -x^2 + 4x - 6$ 21. $f(x) = -2x^2 + 3x + 9$

example 3 determine whether each function has a maximum or minimum value, and find that value. then state the domain and range of the function.

1. $f(x) = 5x^2$ 23. $f(x) = -x^2 - 12$
2. $f(x) = x^2 - 6x + 9$ 25. $f(x) = -x^2 - 7x + 1$
3. $f(x) = 8x - 3x^2 + 2$ 27. $f(x) = 5 - 4x - 2x^2$
4. $f(x) = 15 - 5x^2$ 29. $f(x) = x^2 + 12x + 27$
5. $f(x) = -x^2 + 10x + 30$ 31. $f(x) = 2x^2 - 16x - 42$

Explanation:

Let's solve problem 19: \( f(x) = x^2 - 3x - 10 \) for part a (find y - intercept, axis of symmetry, x - coordinate of vertex), part b (table of values including vertex), and part c (graph the function).

Part a

Step 1: Find the y - intercept

The y - intercept of a function \( y = f(x) \) is found by setting \( x = 0 \).
For \( f(x)=x^{2}-3x - 10 \), when \( x = 0 \), we have \( f(0)=0^{2}-3(0)-10=- 10 \). So the y - intercept is \( (0,-10) \).

Step 2: Find the equation of the axis of symmetry and x - coordinate of the vertex

For a quadratic function in the form \( f(x)=ax^{2}+bx + c \) (here \( a = 1 \), \( b=-3 \), \( c=-10 \)), the formula for the axis of symmetry (and the x - coordinate of the vertex) is \( x=-\frac{b}{2a} \).
Substitute \( a = 1 \) and \( b=-3 \) into the formula: \( x =-\frac{-3}{2(1)}=\frac{3}{2}=1.5 \).
So the equation of the axis of symmetry is \( x = 1.5 \), and the x - coordinate of the vertex is \( x = 1.5 \).

Part b

First, find the y - coordinate of the vertex by substituting \( x = 1.5 \) into \( f(x) \):
\( f(1.5)=(1.5)^{2}-3(1.5)-10=2.25 - 4.5-10=-12.25 \)
So the vertex is \( (1.5,-12.25) \)
Now, we make a table of values. We can choose values of \( x \) around \( x = 1.5 \) (e.g., \( x = 0,1,2,3,4 \)):

\( x \)	\( f(x)=x^{2}-3x - 10 \)
\( 1 \)	\( 1^{2}-3(1)-10=1 - 3-10=-12 \)
\( 1.5 \)	\( - 12.25 \)
\( 2 \)	\( 2^{2}-3(2)-10=4 - 6-10=-12 \)
\( 3 \)	\( 3^{2}-3(3)-10=9 - 9-10=-10 \)
\( 4 \)	\( 4^{2}-3(4)-10=16 - 12-10=-6 \)

Part c

To graph the function:

1. Plot the vertex \( (1.5,-12.25) \).
2. Plot the y - intercept \( (0,-10) \).
3. Plot the other points from the table (e.g., \( (1,-12) \), \( (2,-12) \), \( (3,-10) \), \( (4,-6) \)).
4. Since \( a = 1>0 \), the parabola opens upwards. Draw a smooth curve through the plotted points.

Final Answers (for part a)

• y - intercept: \( \boldsymbol{(0, - 10)} \)
• Axis of symmetry: \( \boldsymbol{x = 1.5} \)
• x - coordinate of vertex: \( \boldsymbol{x = 1.5} \)

(For part b, the table is as shown above. For part c, the graph is a parabola opening upwards with vertex \( (1.5,-12.25) \) and passing through the points in the table)

Answer:

Let's solve problem 19: \( f(x) = x^2 - 3x - 10 \) for part a (find y - intercept, axis of symmetry, x - coordinate of vertex), part b (table of values including vertex), and part c (graph the function).

Part a

Step 1: Find the y - intercept

The y - intercept of a function \( y = f(x) \) is found by setting \( x = 0 \).
For \( f(x)=x^{2}-3x - 10 \), when \( x = 0 \), we have \( f(0)=0^{2}-3(0)-10=- 10 \). So the y - intercept is \( (0,-10) \).

Step 2: Find the equation of the axis of symmetry and x - coordinate of the vertex

For a quadratic function in the form \( f(x)=ax^{2}+bx + c \) (here \( a = 1 \), \( b=-3 \), \( c=-10 \)), the formula for the axis of symmetry (and the x - coordinate of the vertex) is \( x=-\frac{b}{2a} \).
Substitute \( a = 1 \) and \( b=-3 \) into the formula: \( x =-\frac{-3}{2(1)}=\frac{3}{2}=1.5 \).
So the equation of the axis of symmetry is \( x = 1.5 \), and the x - coordinate of the vertex is \( x = 1.5 \).

Part b

First, find the y - coordinate of the vertex by substituting \( x = 1.5 \) into \( f(x) \):
\( f(1.5)=(1.5)^{2}-3(1.5)-10=2.25 - 4.5-10=-12.25 \)
So the vertex is \( (1.5,-12.25) \)
Now, we make a table of values. We can choose values of \( x \) around \( x = 1.5 \) (e.g., \( x = 0,1,2,3,4 \)):

\( x \)	\( f(x)=x^{2}-3x - 10 \)
\( 1 \)	\( 1^{2}-3(1)-10=1 - 3-10=-12 \)
\( 1.5 \)	\( - 12.25 \)
\( 2 \)	\( 2^{2}-3(2)-10=4 - 6-10=-12 \)
\( 3 \)	\( 3^{2}-3(3)-10=9 - 9-10=-10 \)
\( 4 \)	\( 4^{2}-3(4)-10=16 - 12-10=-6 \)

Part c

To graph the function:

1. Plot the vertex \( (1.5,-12.25) \).
2. Plot the y - intercept \( (0,-10) \).
3. Plot the other points from the table (e.g., \( (1,-12) \), \( (2,-12) \), \( (3,-10) \), \( (4,-6) \)).
4. Since \( a = 1>0 \), the parabola opens upwards. Draw a smooth curve through the plotted points.

Final Answers (for part a)

• y - intercept: \( \boldsymbol{(0, - 10)} \)
• Axis of symmetry: \( \boldsymbol{x = 1.5} \)
• x - coordinate of vertex: \( \boldsymbol{x = 1.5} \)

(For part b, the table is as shown above. For part c, the graph is a parabola opening upwards with vertex \( (1.5,-12.25) \) and passing through the points in the table)