Practice Makes Perfect Algebra - Carolyn Wheater [23]
This is a good time to use a decimal approximation, since the x-axis is not usually marked with radicals. So . and You’ll have to estimate those x-intercepts, but even an estimate will help you line up the graph correctly.
The axis of symmetry for y = x2 + 4x + 1 is , so the parabola will be symmetric across the vertical line x = −2. Plug −2 into y = x2 + 4x + 1, and you find that the y-coordinate of the vertex is y =(−2)2 + 4(−2) + 1 = 4 − 8 + 1 = −3. The vertex is(−2, −3).
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EXERCISE 9.5
Find the x- and y-intercepts of each parabola.
1. y = x2 − 4x + 3
2. y = x2 − 4x − 5
3. y = x2 + 2x
4. y = x2 − 7x + 12
5. y = 2x2 + x − 1
Find the axis of symmetry and vertex of each parabola.
6. y = x2 − 8x + 15
7. y = x2 + 4x − 2
8. y = 2x2 − 4x + 3
9. y = − x2 + 6x − 7
10. y = − x2 + 4x + 7
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Table of values
Building a table of values is the most fundamental method of drawing the graph of an equation. Choosing values for x and substituting each one into the equation to find the corresponding value of y will let you plot enough points to graph any equation. When graphing a quadratic equation(or other nonlinear equation), it’s important to make wise choices of x-values. Finding the axis of symmetry, vertex, and intercepts first will tell you where the interesting part of the graph is so that you can choose x-values on both sides of the vertex, but not too far away.
In the previous example, we were getting ready to graph y = x2 + 4x + 1, and we already had four points on the graph. We knew the approximate values of the two x-intercepts, we knew the y-intercept, and we knew the vertex.
Choose a few more values for x on both sides of the vertex to finish your table. Plot the points and connect them with a smooth curve as shown in Figure 9.2.
Figure 9.2 The graph of y = x2 + 4x + 1.
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EXERCISE 9.6
Graph each quadratic function. Use the vertex and intercepts to help make a table of values.
1. y = 2x2 − 1
2. y = − x2 + 8x
3. y = x2 + 2x − 15
4. y = x2 − x − 2
5. y = x2 − 4x + 3
6. y = − x2 + 2x + 5
7. y = x2 + 6x + 9
8. y = 4 − x2
9. y = 2x2 + x − 1
10. y = x2 − 9
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Proportion and variation
A ratio is a comparison of two numbers by division. The relationship between 10 and 5 or between 26 and 13 can be expressed as ratios: 10 : 5 or , both of which are equal to 2 : 1. A proportion is a statement that two ratios are equal, an equation of the form .
Using ratios and extended ratios
If two quantities are in the ratio a: b, it’s not assured that they are exactly equal to a and b, but they are multiples of a and b. As a result, you can represent them as ax and bx and use those expressions to write an equation about the numbers. If two numbers add to 50 and are in ratio 3 : 7, you can represent the numbers as 3x and 7x and write 3x + 7x = 50. You’ll find that x = 5, so the numbers are 3 · 5 and 7 · 5, or 15 and 35.
An extended ratio is a comparison of three or more numbers, usually written in the form a: b: c. If the measurements of the angles of a triangle are in ratio 2 : 3 : 5, you can represent the measures of the angles by 2x, 3x, and 5x and add 2x + 3x + 5x = 180°. Once you solve and find x = 18°, remember to multiply by the appropriate coefficients to find the angle measures: 2x = 36°, 3x = 54°, and 5x = 90°.
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EXERCISE 10.1
Use ratios to solve each problem.
1. Find the number of degrees in each angle of a triangle if the angles are in the ratio 3 : 4 : 5.
2. A piece of wood 20 ft long needs to be cut into two pieces that are in ratio 2 : 3. How long should each piece be?
3. Two numbers are in ratio 8 : 3 and their difference is 65. Find the numbers.
4. Laura determined that the perfect recipe for her raspberry limeade was to mix raspberry juice and lime juice in a 5 : 7 ratio. How much raspberry juice will she need to make 48 oz of the mixture?
5. Three numbers are in ratio 3 : 4 : 8. The sum of the two larger