Math 170 - Cooley Pre-Calculus OCC
Section 3.1 – Quadratic Functions and Inequalities Theorem – Quadratic Functions The graph of any quadratic function is a transformation of the graph of f(x) = x2. Two Forms of Quadratic Functions Parabolic Form: f( x )= a ( x - h )2 + k Quadratic Form: f( x ) = ax2 + bx + c where a 0 . Graphing f ( x )= a ( x – h ) 2 + k The graph of f( x )= a ( x - h )2 + k has the following properties: The vertex is (h , k ) . The axis of symmetry is x= h . For a > 0 , the graph is concave up;. for a < 0 , the graph is concave down. Graphing f ( x )= ax 2 + bx + c The graph of f( x ) = ax2 + bx + c has the following properties: 骣 b骣 b 骣 b4 ac- b2 The vertex is 琪-, f 琪 - , or 琪- , . 桫 2a桫 2 a 桫 2a 4 a b The axis of symmetry is x = - . 2a For a > 0 , the graph is concave up;. for a < 0 , the graph is concave down. Intercepts The x-intercepts and the y-intercept are important points on the graph of a parabola. The x-intercepts are used in solving quadratic inequalities and the y-intercept is the starting point on the graph of a function whose domain is the nonnegative real numbers. The y-intercept is easily found by letting x = 0 and solving for y. The x-intercept(s), if any, are found by letting y = 0 and solving for x. Strategy – Solving a Quadratic Inequality by the Graphical Method 1. Get 0 on one side of the inequality and a quadratic polynomial on the other side. 2. Find all root(s), if any, to the quadratic polynomial. 3. Graph the corresponding quadratic function. The root(s) found in step (2) determine the x-intercept(s) 4. Read the solution set to the inequality from the graph of the parabola. Strategy: Solving a Quadratic Inequality Using Test Points 1. Get 0 on one side of the inequality and a quadratic polynomial on the other side. 2. Find all roots to the quadratic polynomial and plot them on a number line. 3. Select a test point in each interval determined by the roots. 4. Evaluate the quadratic function at each test point. 5. Determine the solution set from the signs of the function at the test points. If there are no real solutions to the quadratic equation that corresponds to a quadratic inequality, then the solution set to the inequality is either the empty set or all real numbers.
1 Math 170 - Cooley Pre-Calculus OCC
Section 3.1 – Quadratic Functions and Inequalities Exercises:
Write each quadratic function in the form y= a( x - h )2 + k and sketch its graph.
1) y= x2 - 6 x
2) y= -2 x2 - 4 x + 8
Find the vertex of the graph of each quadratic function.
1 1 3) y=( x + 6)2 - 4) y= -2 x2 - 8 x + 9 2 4
2 Math 170 - Cooley Pre-Calculus OCC
Section 3.1 – Quadratic Functions and Inequalities Exercises:
Identify the vertex, axis of symmetry, y-intercept, x-intercept(s), if any, and opening (concavity) of each parabola, then sketch the graph.
5) f( x )= ( x + 1)2 - 9
6) y= x2 -6 x + 10
7) y= - x2 +2 x - 6
3 Math 170 - Cooley Pre-Calculus OCC
Section 3.1 – Quadratic Functions and Inequalities Exercises:
Solve each inequality by using the graphical method. State the solution set in interval notation.
8) x2 + x - 2 0 9) x2 -2 x - 1 0
Solve each inequality by using the test-point method. State the solution set in interval notation and graph it.
10) x2 -4 x + 1 0
11) -5 -s2 < 0
4 Math 170 - Cooley Pre-Calculus OCC
Section 3.1 – Quadratic Functions and Inequalities Exercises:
12) If a soldier in basic training fires a rocket-propelled grenade (RPG) straight up from the ground level with an initial velocity of 256 ft/sec, then its height above the ground in feet at time t in seconds is given by the function h( t )= - 16 t2 + 256 t a) What is the maximum height reached by the RPG?
b) How long does it take for the RPG to reach the ground?
13) Chantel wants to make a rectangular frame for a mirror using 10 feet of frame molding. What dimensions will maximize the area of the mirror assuming that there is no waste?
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