3.2E: Describing Relationships with Quadratic Functions (Exercises)
- Page ID
- 99726
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Part 1: Quadratic Functions in Vertex Form
1. Let \(f(x)=3(x-2)^2-1\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is increasing.
- Determine where \(f\) is decreasing.
- Identify the minimum value. Where does it occur?
2. Let \(g(x)=-2(x+1)^2+3\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where\(f\) is increasing.
- Determine where\(f\) is decreasing.
- Identify the maximum value. Where does it occur?
3. Let \(h(x)=0.5(x-2)^2-3\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where\(f\) is increasing.
- Determine where\(f\) is decreasing.
- Identify the minimum value. Where does it occur?
4. Let \(k(x)=-1.2(x+2.1)^2-1.5\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where\(f\) is increasing.
- Determine where\(f\) is decreasing.
- Identify the maximum value. Where does it occur?
5. Let \(f(x)=2(x-1)^2-3\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
6. Let \(f(x)=-3(x+2)^2+4\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
7. Let \(f(x)=0.5(x-3.2)^2+2.5\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
8. Find a formula for the parabola with a
-
Vertex at (4, 0), and \(y\)-intercept (0, -4).
-
Vertex at (5, 6), and \(y\)-intercept (0, -1).
-
Vertex at (-3, 2) that passes through the point (3, -2).
-
Vertex at (1, -3) that passes through the point (-2, 3).
9. Find a formula for the parabola whose graph is given below
10. A ball is kicked into the air. The height of the ball (in feet) after t seconds from where is was kicked is given by \(h=f(t)=-16(t-3)^2+50 \).
- What was the maximum height of the ball?
- When did the ball reach its maximum height?
Part 2: Quadratic functions in Standard Form
11. Let \(f(x)=3x^{2} +6x-9\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is increasing.
- Determine where \(f\) is decreasing.
- Identify the minimum value. Where does it occur?
12. Let \(f(x)=2x^{2} -10x+4\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is increasing.
- Determine where \(f\) is decreasing.
- Identify the minimum value. Where does it occur?
13. \(h(t)=-4t^{2} +6t-1\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(h\) is increasing.
- Determine where \(h\) is decreasing.
- Identify the maximum value. Where does it occur?
14. \(k(x)=2x^{2} +4x-15\)
- Identify the vertex of the parabola.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(k\) is increasing.
- Determine where \(k\) is decreasing.
- Identify the minimum value. Where does it occur?
15. Let \(f(x)=x^2-6x+8\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
16. Let \(f(x)=2x^2-5x-3\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
17. Let \(f(x)=3.2x^2-4.2x-1.3\)
- Determine the zeros.
- Determine the end behavior.
- Sketch a graph of the parabola.
- Determine where \(f\) is positive.
- Determine where \(f\) is negative.
18. Write an equation for a quadratic with the given features
- Zeros at \(x = -3\) and \(x = 1\) with a \(y\)-intercept (0, 2)
- Zeros at \(x = 2\) and \(x = -5\) with a \(y\)-intercept (0, 3)
- Zeros at \(x = 2\) and \(x = 5\) that passes through the point (-1,4).
- Zeros at \(x = -1.2\) and \(x = -2,3\) that passes through the point (2,5).
19. A rocket is launched in the air. Its height above sea level (in meters) in t seconds is given by \(h(t)=-4.9t^{2} +229t+234\).
a. From what height was the rocket launched?
b. How high above sea level does the rocket reach its peak?
c. Assuming the rocket will splash down in the ocean, at what time does splashdown occur?
20. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by \(h(t)=-4.9t^{2} +24t+8\).
a. From what height was the ball thrown?
b. How high above ground does the ball reach its peak?
c. When does the ball hit the ground?
21. A javelin is thrown in the air. Its height is given by \(h(x)=-\dfrac{1}{20} x^{2} +8x+6\), where x is the horizontal distance in feet from the point at which the javelin is thrown.
a. How high is the javelin when it was thrown?
b. What is the maximum height of the javelin?
c. How far from the thrower does the javelin strike the ground?
- Answer
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1. a) Vertex at (2,-1) b) The graph opens up on both ends (as \(x \to \pm \infty, y \to \infty ) \)
c)
d) \(f\) is increasing on \( (2,\infty) \) e) \(f\) is decreasing on \((-\infty,2) \) f) The minimum value is \( y = -1\) and occurs at \(x=2\)
5. a) Zeros at \(x = 1 \pm \sqrt{\dfrac{3}{2}}\) or \(x \approx -0.2247 \text{ and } x \approx 2.2247 \) b) The graph opens up on both ends (as \(x \to \pm \infty, y \to \infty ) \)
c)
d) \(f\) is positive on \((-\infty,-0.2247 ) \bigcup (2.2247,\infty ) \) e) \(f\) is negative on \( (-0.2247,2.2247 ) \).
8. a) \(y= \dfrac{-1}{4}(x-4)^2 \)
9. a) \(y=3(x-3)^2+4 \)
11. a) Vertex at (-1,-12) b) The graph opens up on both ends (as \(x \to \pm \infty, y \to \infty ) \)
c)
d) \(f\) is increasing on \( (-1,\infty) \) e) \(f\) is decreasing on \(-\infty,-1)\) f) The minimum value is \( y = -12\) . It occurs at \(x=-1\)
15. a) The zeros are \(x=2\) and \(x=4\). b) The graph opens up on both ends (as \(x \to \pm \infty, y \to \infty ) \)
c)
d) \(f\) is positive on \((-\infty,2 ) \bigcup (4,\infty ) \) e) \(f\) is negative on \( (2,4 ) \).
18. a) \(y=\dfrac{-2}{3}(x+3)(x-1) \)