11.1: Graphing Quadratic Equations
This activity is intended to supplement Algebra I, Chapter 10, Lesson 1.
In this activity, you will explore:
- Graphing quadratic equations
- Comparing the value of constants in the equations to the coordinates and axes of symmetry on the graph
Before beginning this activity, clear out any functions from the \begin{align*}Y=\end{align*} screen and turn all plots off.
Problem 1 – Vertex Form
Enter the equation \begin{align*}y = x^2\end{align*} into \begin{align*}Y1\end{align*}. Press ZOOM and select ZStandard to view the graph in a standard size window.
1. Describe the shape of the curve, which is called a parabola.
The vertex form of a parabola is \begin{align*}y = a(x - h)^2 + k\end{align*}.
For example, the equation \begin{align*}y = 2(x - 3)^2 + 1\end{align*} is in vertex form. Graph this equation in \begin{align*}Y1\end{align*}.
2. What is the value of \begin{align*}a\end{align*}?
3. What is the value of \begin{align*}h\end{align*}?
4. What is the value of \begin{align*}k\end{align*}?
Next we will use the TI-84 application Transformation Graphing to see how the values of \begin{align*}a, h\end{align*}, and \begin{align*}k\end{align*} affect the characteristics of the parabola (such as the vertex, axis of symmetry, and maximum or minimum values).
To open the Transformation Graphing app, press APPS then Transfrm from the menu.
Go to the \begin{align*}Y=\end{align*} screen and enter \begin{align*}A(X - B)^2+ C\end{align*} in \begin{align*}Y1\end{align*}. This is the equation of a parabola in vertex form. The Transformation Graphing application requires that we use the variables \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*} instead of \begin{align*}a, h\end{align*}, and \begin{align*}k\end{align*}.
Press GRAPH. The calculator has chosen values for \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*} and graphed a parabola. Note that the \begin{align*}=\end{align*} next to A is highlighted.
Press the down arrow to move to the \begin{align*}=\end{align*} next to \begin{align*}B\end{align*}. Remember that \begin{align*}B\end{align*} corresponds to \begin{align*}h\end{align*} in the vertex form \begin{align*}y = a(x - h)^2 + k\end{align*}.
Change the value of \begin{align*}B (h)\end{align*} and observe the effect on the graph. You can type in a new value and press \begin{align*}e\end{align*} or use the left and right arrow keys to decrease or increase the value of \begin{align*}B\end{align*} by \begin{align*}1\end{align*}.
5. What happens when \begin{align*}h\end{align*} is positive?
6. What happens when \begin{align*}h\end{align*} is negative?
7. What happens as the absolute value of \begin{align*}h\end{align*} gets larger?
8. What happens as the absolute value of \begin{align*}h\end{align*} gets smaller?
9. What do you think will happen to the parabola if \begin{align*}h\end{align*} is ?
10. Change \begin{align*}h\end{align*} to zero. Was your hypothesis correct?
11. Record the equation of your parabola.
\begin{align*}a = A = \underline{\;\;\;\;\;\;\;\;}\quad h = B = 0 \quad k = C =\underline{\;\;\;\;\;\;\;\;} \quad y = a(x - h)^2 + k =\underline{\;\;\;\;\;\;\;\;} (x - 0)^2 + \underline{\;\;\;\;\;\;\;\;}\end{align*}
We will now turn our attention to a different feature of the graph of a quadratic function: the axis of symmetry. First, turn off the Transformation Graphing app. Go to Apps \begin{align*}>\end{align*} Transfrm \begin{align*}>\end{align*} Uninstall.
Next, enter the equation you recorded in question 11 in \begin{align*}Y1\end{align*}.
Press GRAPH to view its graph.
Now we would like to draw a line parallel to the \begin{align*}x-\end{align*}axis that intersects the parabola twice, as in the graph shown. Experiment with different equations in \begin{align*}Y2\end{align*} until you find such a line.
Record the equation of the line. ______________
Use the intersect command to find the coordinates of the two points where the line intersects the parabola. Press \begin{align*}2^{nd}\end{align*} [TRACE] to open the Calculate menu, choose intersect, and follow the prompts. Record the coordinates of the intersection in the table on the next page.
Line | Left intersection | Distance from left intersection to \begin{align*}y-\end{align*}axis | Right intersection | Distance from right intersection to \begin{align*}y-\end{align*}axis |
---|---|---|---|---|
\begin{align*}y =\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} | ||
\begin{align*}y =\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} | ||
\begin{align*}y =\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} | \begin{align*}(\;\;\;\;\;\;,\;\;\;\;\;)\end{align*} |
Choose a new line parallel to the \begin{align*}x-\end{align*}axis and find the coordinates of its intersection with the parabola. Repeat several times, recording the results in the table above.
12. What do you notice about the points in the table? How do their \begin{align*}x-\end{align*}coordinates compare? How do their \begin{align*}y-\end{align*}coordinates compare? Calculate the distance from each intersection point to the \begin{align*}y-\end{align*}axis.
13. What do you notices about the distances from each intersection point to the \begin{align*}y-\end{align*}axis?
The relationships you see exist because the graph is symmetric and the \begin{align*}y-\end{align*}axis is the axis of symmetry.
14. What is the equation of the axis of symmetry?
How do you think the graph will move if \begin{align*}h\end{align*} is changed from to \begin{align*}4\end{align*}? Change the value of \begin{align*}h\end{align*} in the equation in \begin{align*}Y1\end{align*} from to \begin{align*}4\end{align*}. \begin{align*}Y1 = (X-4)^2 - 2\end{align*}
As before, enter an equation in \begin{align*}Y2\end{align*} to draw a line parallel to the \begin{align*}x-\end{align*}axis that passes through the parabola twice, as shown. Find the two intersection points.
Left intersection: _____________
Right intersection: ______________
The axis of symmetry runs through the midpoint of these two points. Use the formula to find the midpoint of the two intersection points.
midpoint: ______________
midpoint \begin{align*}(x1, y1)\end{align*} and \begin{align*}(x2, y2) = \left (\frac{x1+x2}{2}, \frac{y1+y2}{2} \right )\end{align*}
Draw a vertical line through this midpoint. Press \begin{align*}2^{nd}\end{align*} [MODE] to return to the home screen. Then press \begin{align*}2^{nd}\end{align*} [PRGM] to open the Draw menu, choose the Vertical command, and enter the \begin{align*}x-\end{align*}coordinate of the midpoint. The command shown here draws a vertical line at \begin{align*}x = 4\end{align*}.
This vertical line the axis of symmetry.
Use the Trace feature to approximate the coordinates of the point where the vertical line intersects the parabola. Round your answer to the nearest tenth. This point is the vertex of the parabola. vertex: ______________
15. Look back at the equation in \begin{align*}Y1\end{align*}. How is the vertex related to the general equation \begin{align*}y = a(x - h)^2 + k\end{align*}?
Since the vertex is the lowest point on the graph, it is also the minimum. Check your answer by pressing \begin{align*}2^{nd}\end{align*} [TRACE] for the Calculate menu and select the minimum command.
Now we will examine the effect of the value of a on the “width” of the parabola. Turn the Transformation Graphing app on again and enter \begin{align*}A(X - B)^2 + C\end{align*} in \begin{align*}Y1\end{align*}.
Change the value of \begin{align*}A\ (a)\end{align*} and observe the effect on the graph. You can type in a new value and press \begin{align*}e\end{align*} or use the left and right arrow keys to decrease or increase the value of \begin{align*}A\end{align*} by \begin{align*}1\end{align*}.
16. What happens when \begin{align*}a\end{align*} is positive?
17. What happens when \begin{align*}a\end{align*} is negative?
18. What happens as the absolute value of \begin{align*}a\end{align*} gets larger?
19. What happens as the absolute value of \begin{align*}a\end{align*} gets smaller?
If the graph opens downward (\begin{align*}a\end{align*} is negative), the vertex is a maximum because it is the highest point on the graph.
The vertex form of a parabola is \begin{align*}y = a(x - h)^2 + k\end{align*}.
20. The coefficient ____ determines whether the parabola opens upward or downward, and how wide the parabola is.
21. The vertex of the parabola is the point with coordinates _______.
22. The equation of the axis of symmetry is \begin{align*}x = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}.
Sketch the graph of each function. Then check your graphs with your calculator. (Turn off Transformation Graphing first.)
23. \begin{align*}y = x^2 - 3\end{align*}
24. \begin{align*}y = (x-7)^2\end{align*}
25. \begin{align*}y = - (x+5)^2+4\end{align*}
Problem 2 – Standard Form
The standard form of a parabola is \begin{align*}y = ax^2 + bx + c\end{align*}. Let’s see how the standard form relates to the vertex form.
\begin{align*}y & = a(x-h)^2+k \\ y & = a(x^2 - 2xh + h^2)+k && b = -2ah \\ y & = \begin{array} {|c|} \hline a\\ \hline \end{array}x^2 \begin{array} {|c|} \hline -2ah\\ \hline \end{array}x+ \begin{array} {|c|} \hline ah^2+k\\ \hline \end{array} && h = - \frac{b}{2a} \\ y & = \begin{array} {|c|} \hline a\\ \hline \end{array}x^2 + \begin{array} {|c|} \hline b\\ \hline \end{array}x+\begin{array} {|c|} \hline c\\ \hline \end{array}\end{align*}1. For the standard form of a parabola \begin{align*}y = ax^2 + bx + c\end{align*}, the \begin{align*}x-\end{align*}coordinate of the vertex is _____.
The equation \begin{align*}y = 2x^2 - 4\end{align*} is in standard form. Graph this equation in \begin{align*}Y1\end{align*}.
2. What is the value of \begin{align*}a\end{align*}?
3. What is the value of \begin{align*}b\end{align*}?
4. What is the value of \begin{align*}c\end{align*}?
5. What is the \begin{align*}x-\end{align*}coordinate of the vertex?
6. Use the minimum command to find the vertex of the parabola.
vertex: _____________
How do you think changing the coefficient of \begin{align*}x^2\end{align*} might affect the parabola? Begin by turning on the Transformation Graphing app. (APPS, scroll down to TRANSFRM)
Enter the equation for the standard form of a parabola in \begin{align*}Y1\end{align*}.
Try different values of \begin{align*}A\end{align*} in the equation. You can type in a new value and press ENTER or use the left and right arrow keys to decrease or increase the value of \begin{align*}A\end{align*} by \begin{align*}1\end{align*}.
Make sure to test values of \begin{align*}A\end{align*} that are between \begin{align*}-1\end{align*} and \begin{align*}1\end{align*}. To do this, you can type in a value, as in the screenshot shown.
You can also adjust the size of the increase and decrease when you use the right and left arrows. Press WINDOW and arrow over to Settings. Then change the value of the step to \begin{align*}0.1\end{align*} or another value less than \begin{align*}1\end{align*}.
7. Does the value of a change the position of the vertex?
8. How does the value of a related to the shape of the parabola?
Next we will explore another feature of the parabola: the \begin{align*}y-\end{align*}intercept. To find the \begin{align*}y-\end{align*}intercept of the parabola, use the value feature, found in the Calculate menu, to find the value of the equation at \begin{align*}x = 0\end{align*}.
Change the values of \begin{align*}a, b\end{align*}, and/or \begin{align*}c\end{align*} and find the \begin{align*}y-\end{align*}intercept. Repeat several times and record the results in the table below.
Equation | \begin{align*}a\end{align*} | \begin{align*}b\end{align*} | \begin{align*}c\end{align*} | \begin{align*}y-\end{align*}intercept |
---|---|---|---|---|
\begin{align*}y = 2x^2 - 4\end{align*} | \begin{align*}2\end{align*} | \begin{align*}-4\end{align*} | \begin{align*}-4\end{align*} |
9. How does the equation of the parabola in standard form relate to the \begin{align*}y-\end{align*}intercept of the parabola?
Sketch the graph of each function. Then check your graphs with your calculator. (Turn off Transformation Graphing first.)
10. \begin{align*}y = x^2 + 6x + 2\end{align*}
11. \begin{align*}y = -x^2-4x\end{align*}
12. \begin{align*}y = -2x^2 + 8x+5\end{align*}
SUMMARY OF GRAPHING QUADRATICS
1. The vertex form of a parabola is __________________.
2. The coefficient ____ determines whether the parabola opens upward or downward, and how wide the parabola is. The vertex of the parabola is the point with coordinates _______. The equation of the axis of symmetry is \begin{align*}x =\end{align*} ______.
3. The standard form of a parabola is ____________________.
4. The \begin{align*}x-\end{align*}coordinate of the vertex is ______________. The equation of the axis of symmetry is \begin{align*}x =\end{align*} ______. The \begin{align*}y-\end{align*}intercept is ________.
Sketch the graph of each function. Identify the vertex and the equation of the axis of symmetry. Then check your graphs with your calculator.
5. \begin{align*}y = x^2 + 4\end{align*}
vertex _____
axis of symmetry _____
6. \begin{align*}y = (x-3)^2+5\end{align*}
vertex _____
axis of symmetry _____
7. \begin{align*}y = -(x-2)^2\end{align*}
vertex _____
axis of symmetry _____
8. \begin{align*}y = x^2+6x+9\end{align*}
vertex _____
axis of symmetry _____
9. \begin{align*}y = -3x^2+6x+1\end{align*}
vertex _____
axis of symmetry _____
10. \begin{align*}y = x^2+1\end{align*}
vertex _____
axis of symmetry _____
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