Preview Warm Up California Standards Lesson Presentation
Warm Up 1. Solve 2x – 3y = 12 for y. 2. Evaluate the function f(x) = for –10, –5, 0, 5, and 10. –1 f(–10) = 0 f(–5) = 1 f(0) = 2 f(5) = f(10) = 3
Vocabulary linear equation linear function
California Standards 6.0 Students graph a linear equation and compute x- and y- intercepts (e.g. graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequalities (e.g., they sketch the region defined by 2x + 6y < 4). Also covered: 7.0, 17.0, 18.0
Many stretches on the German autobahn have no speed limit. If a car travels continuously at 120 km/hr, y = 120x gives the number of kilometers y that the car would travel in x hours. Notice that the graph is a straight line. An equation whose graph forms a straight line is a linear equation. Also notice that this is a function. A function represented by a linear equation is a linear function.
For any two points, there is exactly one line that contains them both. This means you need only two ordered pairs to graph a line. However, graphing three points is a good way to check that your line is correct.
Graphing Linear Equations Graph y = 2x + 1. Tell whether it represents a function. Step 1 Choose three values of x and generate ordered pairs. (1, 3) y = 2(1) + 1 = 3 1 y = 2(0) + 1 = 1 (0, 1) 0 –1 y = 2(–1) + 1 = –1 (–1, –1) Step 2 Plot the points and connect them with a straight line. No vertical line will intersect this graph more than once. So y = 2x + 1 describes a function.
Helpful Hint Sometimes solving for y first makes it easier to generate ordered pairs using valuesof x. To review solving for a variable, see Lesson 2-6.
Special Linear Equation-A Graph x =–2. Tell whether it represents a function. Any ordered pair with an x-coordinate of –2 will satisfy this equation. Plot several points that have an x-coordinate of –2 and connect them with a straight line. There is a vertical line that intersects this graph more than once, so x = –2 does not represent a function.
Special Linear Equation-A Graph y =8. Tell whether it represents a function. Any ordered pair with a y-coordinate of 8 will satisfy this equation. Plot several points that have an y-coordinate of 8 and connect them with a straight line. No vertical line will intersect this graph more than once, so y = 8 represents a function.
Linear equations can be written in the standard form as shown below.
Additional Example 3B: Writing Linear Equations in Standard Form Write x = 4in standard form and give the values of A, B, and C. Then describe the graph. x = 4 The equation is in standard form. x + 0y = 4 A = 1, B = 0, C = 4 The graph is a vertical line at x = 4.
y = 5x – 9 –5x –5x –5x + y = – 9 Check It Out! Example 3a Write y = 5x – 9 in standard form and give the values of A, B, and C. Then describe the graph. Subtract 5x from both sides. The equation is in standard form. A = –5, B = 1, C = –9 The graph is a line that is neither horizontal nor vertical.
Linear Equations
Linear equations can be written in the standard form as shown below.
Linear Equations: • Equations whose graphs are lines are Linear equations. • An equation is linear if • the variables occur as first powers only • There are no products of variables • No variables in the denominator. • The Equation form iswhere A, B, and C are Integers.
Review Forms of Linear Equations • Standard Form: • Slope-Intercept Form: • Point Slope Form:
Graphing Using Intercepts • We can graph a linear equations by finding any two points on the graph. • Often the easiest points to find are the points where the graph crosses the axes. • X-intercepts: where y=0 • Y-intercepts: where x=0
Graphing horizontal and Vertical lines. • Standard form of a linear equation: • Graph y = 3 • Graph x = -4
7.3 Exercises (Graph using 3-points) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)
7.3 Exercises (Graph) • x – 1 = y • x – 3 = y • 2x – 1 = y • 3x – 2 = y • 4x – 3y = 12 • 6x – 2y = 18 • 7x + 2y = 6 • 3x + 4y = 5 • y = – 4 – 4x • Y = – 3 – 3x
Lesson Quiz: Part I Graph each linear equation. Then tell whether it represents a function. 1. 2y + x = 6 2. 3y = 12 Yes, it is a function. Yes, it is a function.
Lesson Quiz: Part II Without graphing, tell whether each point is on the graph of 6x – 2y = 8. yes 4. (3, 5) 3. (1, 1) no 5. The cost of a can of iced-tea mix at SaveMore Grocery is $4.75. The function f(x) = 4.75x gives the cost of x cans of iced-tea mix. Graph this function and give its domain and range. D: {0, 1, 2, 3, …} R: {0, 4.75, 9.50, 14.25, …}
