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Use Graphs to Solve Quadratic Equations

Identify x-intercepts of parabolas

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Use Graphs to Solve Quadratic Equations

Suppose a batter hits a baseball, and the height of the baseball above the ground \begin{align*}t\end{align*} seconds after it is hit can be modeled by the function \begin{align*}h(t)=-16t^2 + 50t + 2\end{align*}. What is the maximum height that the baseball would attain? How long would it take for the baseball to hit the ground? 

Using Graphs to Solve Quadratic Equations

Projectile Motion

Quadratic equations are used to describe projectile motion, or the motion of an object thrown into the air with both horizontal and vertical movement. Think of a baseball being hit by a batter. 

Isaac Newton’s theory for projectile motion is represented by the equation:

\begin{align*}h(t)=- \frac{1}{2} (g) t^2+v_0 t+h_0\end{align*}

  • \begin{align*}t=\end{align*}time (usually in seconds)
  • \begin{align*}g=\end{align*}gravity due to acceleration; either 9.8 m/s or 32 ft/s
  • \begin{align*}v_0=\end{align*}initial velocity
  • \begin{align*}h_0=\end{align*}initial height of object

Using the equation from above, write an equation for the following situation and describe the shape of the graph:

A quarterback throws a football at an initial height of 5.5 feet with an initial velocity of 35 feet per second. 

Substitute the appropriate information into the equation for projectile motion:

  • \begin{align*}g=32\end{align*} because the information is given in feet
  • \begin{align*}v_0=35\end{align*}
  • \begin{align*}h_0=5.5\end{align*}

The equation becomes \begin{align*}h(t)=-\frac{1}{2} (32) t^2+35t+5.5 \rightarrow h(t)=-16t^2+35t+5.5\end{align*}.

Using the information from the previous Concept, we know:

  • The value of \begin{align*}a\end{align*} is negative, so the parabola opens downward.
  • The vertex is a maximum point.
  • The \begin{align*}y-\end{align*}intercept is (0, 5.5).
  • The graph is narrow about its line of symmetry.

Solving a Quadratic Equation Using a Graphing Calculator

Problems can with quadratic equations be solved using graphs of equations. We can also use a graphing calculator to make this easier.

Using the information from above about the football, let's determine at what time the football will be 6 feet high.

Previous Concepts focused on how to solve systems by graphing. We will be using similar ideas to find at what point the function \begin{align*}h(t)=-16t^2+35t+5.5\end{align*} is equal to 6. You can think of this situation as a system: \begin{align*}\begin{cases} y=-16t^2+35t+5.5\\ y=6 \end{cases}\end{align*}. You are looking for the appropriate \begin{align*}x-\end{align*}coordinates that give a \begin{align*}y-\end{align*}coordinate of 6 feet. Therefore, you are looking for the intersection of the two equations.

Begin by typing the equations into the \begin{align*}[Y=]\end{align*} menu of your calculator. Adjust the window until you see the vertex, \begin{align*}y-\end{align*}intercept, \begin{align*}x-\end{align*}intercepts, and the horizontal line of 6 units.

By looking at the graph, you can see there are two points of intersection. To find the point of intersection, press [2ND] and then [TRACE] to get to the [CALC] menu. Select [INTERSECT]. Now, use the up and down arrows to get close to one of the intersections. Press [ENTER] and the pointer will move to the second equation. Make sure the pointer is close to that same intersection and press [ENTER] again. Press [ENTER] one more time and the intersection will be displayed on the bottom of the screen. To find the second intersection, repeat the steps but move the pointer close to the other intersection point.

The intersections are:

\begin{align*}(0.014,6) \ and \ (2.172,6)\end{align*}

At 0.014 seconds and again at 2.17 seconds, the football is six feet from the ground.

Using a Calculator to Find the Vertex

You can also use a graphing calculator to determine the vertex of the parabola. The vertex of this equation is a maximum point, so in the [CALC] menu of the graphing option, look for [MAXIMUM].

Choose option #4. The calculator will ask you, “LEFT BOUND?” Move the cursor to the left of the vertex and hit [ENTER].

The calculator will ask, “RIGHT BOUND?” Move the cursor to the right of the vertex and hit [ENTER].

Hit [ENTER] to guess.

Now, let's use the information from the football problem above and a graphing calculator to complete the following problem:

When will the football reach its maximum height? When will it reach 25 feet high? When will the football hit the ground, assuming no one will catch it?

Using the steps above, find the maximum. The maximum point on this parabola is (1.09, 24.64).

The vertex represents the maximum point of this quadratic equation. Since its height is 24.64 feet, we can safely say the football will not reach 25 feet.

We now want to know at what time the height is zero: \begin{align*}\begin{cases} y=-16t^2+35t+5.5 \\ y=0 \end{cases}\end{align*}. By repeating the process above and finding the intersection of the two lines, the solution is (2.33, 0). At 2.33 seconds, the ball will hit the ground.

The point at which the ball reaches the ground \begin{align*}(y=0)\end{align*} represents the \begin{align*}x-\end{align*}intercept of the graph.

The Number of Solutions for a Quadratic Equation

The \begin{align*}x-\end{align*}intercept of a quadratic equation is also called a root, solution, or zero. Quadratic equations have zero solutions when they never cross the \begin{align*}x\end{align*}-axis, one solution when the vertex is the only point that intersects the \begin{align*}x\end{align*}-axis, and two solutions, or two \begin{align*}x-\end{align*}intercepts, in any other case.

Let's determine the number of solutions to \begin{align*}y=x^2+4\end{align*}:

Graph this quadratic equation, either by hand or with a graphing calculator. Adjust the calculator’s window to see both halves of the parabola, the vertex, the \begin{align*}x-\end{align*}axis, and the \begin{align*}y-\end{align*}intercept.

This parabola does not cross the \begin{align*}x-\end{align*}axis. The solutions to a quadratic equation are its \begin{align*}x-\end{align*}intercepts and since this does not have any, this equation has no real solutions.

   

 

Examples

Example 1

Earlier, you were told that the height of a baseball above the ground \begin{align*}t\end{align*} seconds after it is hit can be modeled by the function \begin{align*}h(t)=-16t^2 + 50t + 2\end{align*}. What is the maximum height that the baseball will attain? How long would it take for the baseball to hit the ground?

To find the maximum height of the baseball, you need to find the maximum of the given function. Using the steps from the Using a Calculator to Find the Vertex section, use a graphing calculator to find the value. 

The vertex is at the point (1.563, 41.063). Therefore, the maximum height that the baseball will attain is 41.063, the \begin{align*}y\end{align*}-value of the vertex.

The baseball will hit the ground at the \begin{align*}x\end{align*}-intercepts, if they exist.   In other words, we want to solve the system \begin{align*}\begin{cases} y=-16t^2+50t+2 \\ y=0 \end{cases}\end{align*}.

Using a calculator to find the intersection of these two graphs as in the Solving a Quadratic Equation Using a Graphing Calculator section, we get the \begin{align*}x\end{align*}-intercepts to be (3.165, 0) and (-0.04, 0). We cannot have negative time so it takes 3.165 seconds to hit the ground.

Example 2

Andrew has 100 feet of fence to enclose a rectangular tomato patch. He wants to find the dimensions of the rectangle that encloses the most area.

The perimeter of a rectangle is the sum of all four sides. Let \begin{align*}w=\end{align*} width and \begin{align*}l=\end{align*} length. The perimeter of the tomato patch is \begin{align*}100=l+l+w+w \rightarrow 100=2l+2w\end{align*}.

The area of a rectangle is found by the formula \begin{align*}A=l(w)\end{align*}. We are looking for the intersection between the area and perimeter of the rectangular tomato patch. This is a system.

\begin{align*}\begin{cases} 100 = 2l+2w \\ A = l(w) \end{cases}\end{align*}

Before we can graph this system, we need to rewrite the first equation for either \begin{align*}l\end{align*} or \begin{align*}w\end{align*}. We will then use the Substitution Property.

\begin{align*}100 &= 2l+2w \rightarrow 100-2l=2w\\ \frac{100-2l}{2} &= w \rightarrow 50-l=w\end{align*}

Use the Substitution Property to replace the variable \begin{align*}w\end{align*} in the second equation with the expression \begin{align*}50-l\end{align*}.

\begin{align*}A=l(50-l)=50l-l^2\end{align*}

Graph this equation to visualize it.

The parabola opens downward so the vertex is a maximum. The maximum value is (25, 625). The length of the tomato patch should be 25 feet long to achieve a maximum area of 625 square feet.

Review

  1. What are the alternate names for the solution to a parabola?
  2. Define the following variables in the function \begin{align*}h(t)=-\frac{1}{2} (g) t^2+v_0 t+h_0\end{align*}.
    1. \begin{align*}h_0\end{align*}
    2. \begin{align*}t\end{align*}
    3. \begin{align*}v_0\end{align*}
    4. \begin{align*}g\end{align*}
    5. \begin{align*}h(t)\end{align*}
  1. A rocket is launched from a height of 3 meters with an initial velocity of 15 meters per second.
    1. Model the situation with a quadratic equation.
    2. What is the maximum height of the rocket? When will this occur?
    3. What is the height of the rocket after four seconds? What does this mean?
    4. When will the rocket hit the ground?
    5. At what time will the rocket be 13 meters from the ground?

How many solutions does the quadratic equation have?

  1. \begin{align*}-x^2+3=0\end{align*}
  2. \begin{align*}2x^2+5x-7=0\end{align*}
  3. \begin{align*}-x^2+x-3=0\end{align*}

Find the zeros of the quadratic equations below. If necessary, round your answers to the nearest hundredth.

  1. \begin{align*}y=-x^2+4x-4\end{align*}
  2. \begin{align*}y=3x^2-5x\end{align*}
  3. \begin{align*}x^2+3x+6=0\end{align*}
  4. \begin{align*}-2x^2+x+4=0\end{align*}
  5. \begin{align*}x^2-9=0\end{align*}
  6. \begin{align*}x^2+6x+9=0\end{align*}
  7. \begin{align*}10x^2-3x^2=0\end{align*}
  8. \begin{align*}\frac{1}{2}x^2-2x+3=0\end{align*}
  9. \begin{align*}y=-3x^2+4x-1\end{align*}
  10. \begin{align*}y=9-4x^2\end{align*}
  11. \begin{align*}y=x^2+7x+2\end{align*}
  12. \begin{align*}y=-x^2-10x-25\end{align*}
  13. \begin{align*}y=2x^2-3x\end{align*}
  14. \begin{align*}y=x^2-2x+5\end{align*}
  15. Andrew is an avid archer. He launches an arrow that takes a parabolic path, modeled by the equation \begin{align*}y=-4.9t^2+48t\end{align*}. Find how long it takes the arrow to come back to the ground.

For questions 24 – 26,

(a) Find the roots of the quadratic polynomial.

(b) Find the vertex of the quadratic polynomial.

  1. \begin{align*}y=x^2+12x+5\end{align*}
  2. \begin{align*}y=x^2+3x+6\end{align*}
  3. \begin{align*}y=-x^2-3x+9\end{align*}
  4. Sharon needs to create a fence for her new puppy. She purchased 40 feet of fencing to enclose three sides of a fence. What dimensions will produce the greatest area for her puppy to play?
  5. An object is dropped from the top of a 100-foot-tall building.
    1. Write an equation to model this situation.
    2. What is the height of the object after 1 second?
    3. What is the maximum height of the object?
    4. At what time will the object be 50 feet from the ground?
    5. When will the object hit the ground?

Mixed Review

  1. Factor \begin{align*}3r^2-4r+1\end{align*}.
  2. Simplify \begin{align*}(2+\sqrt{3})(4+\sqrt{3})\end{align*}.
  3. Write the equation in slope-intercept form and identify the slope and \begin{align*}y-\end{align*}intercept: \begin{align*}9-3x+18y=0\end{align*}.
  4. The half life of a particular substance is 16 days. An organism has 100% of the substance on day zero. What is the percentage remaining after 44 days?
  5. Multiply and write your answer in scientific notation: \begin{align*}0.00000009865 \times 123564.21\end{align*}.
  6. A mixture of 12% chlorine is mixed with a second mixture containing 30% chlorine. How much of the 12% mixture is needed to mix with 80 mL to make a final solution of 150 mL with a 20% chlorine concentration?

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.3. 

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Vocabulary

solutions of a quadratic equation

The x-intercept of a quadratic equation is also called a root, solution, or zero.

Double Root

A solution that is repeated twice.

Quadratic Formula

The quadratic formula states that for any quadratic equation in the form ax^2+bx+c=0, x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.

Roots

The roots of a function are the values of x that make y equal to zero.

Zero Product Rule

The zero product rule states that if the product of two expressions is equal to zero, then at least one of the original expressions much be equal to zero.

Zeroes

The zeroes of a function f(x) are the values of x that cause f(x) to be equal to zero.

Zeros

The zeros of a function f(x) are the values of x that cause f(x) to be equal to zero.

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