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Find the intercepts of the parabola y = 9 x 2 + 12 x + 4 .

y : ( 0 , 4 ) ; x : ( 2 3 , 0 )

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Graph quadratic equations in two variables

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.

How to graph a quadratic equation in two variables

Graph y = x 2 6 x + 8 .

Solution

The image shows the steps to graph the quadratic equation y equals x squared minus 6 x plus 8. Step 1 is to write the quadratic equation with y on one side. This equation has y on one side already. The value of a is one, the value of b is -6 and the value of c is 8. Step 2 is to determine whether the parabola opens upward or downward. Since a is positive, the parabola opens upward. Step 3 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the quantity 2 a. Plugging in the values of b and a the formula becomes x equals negative -6 divided by the quantity 2 times 1 which simplifies to x equals 3. The axis of symmetry is the line x equals 3. Step 4 is to find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the equation and solve for y. The equation is y equals x squared minus 6 x plus 8. Replacing x with 3 it becomes y equals 3 squared minus 6 times 3 plus 8 which simplifies to y equals -1. The vertex is (3, -1). Step 5 is to find the y-intercept and find the point symmetric to the y-intercept across the axis of symmetry. We substitute x equals 0 into the equation. The equation is y equals x squared minus 6 x plus 8. Replacing x with 0 it becomes y equals 0 squared minus 6 times 0 plus 8 which simplifies to y equals 8. The y-intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. The point symmetric to the y-intercept is (6, 8). Step 6 is to find the x-intercepts. We substitute y equals 0 into the equation. The equation becomes 0 equals x squared minus 6 x plus 8. We can solve this quadratic equation by factoring to get 0 equals the quantity x minus 2 times the quantity x minus 4. Solve each equation to get x equals 2 and x equals 4. The x-intercepts are (2, 0) and (4, 0). Step 7 is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these five points to sketch the parabola. The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -2 to 10. The y-axis of the plane runs from -3 to 10. The vertex is at the point (3, -1). Four points are plotted on the curve at (0, 8), (6, 8), (2, 0) and (4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3.
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Graph the parabola y = x 2 + 2 x 8 .

y : ( 0 , −8 ) ; x : ( 2 , 0 ) , ( −4 , 0 ) ;
axis: x = −1 ; vertex: ( −1 , −9 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, -9). Three points are plotted on the curve at (0, -8), (2, 0) and (-4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

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Graph the parabola y = x 2 8 x + 12 .

y : ( 0 , 12 ) ; x : ( 2 , 0 ) , ( 6 , 0 ) ;
axis: x = 4 ; vertex: ( 4 , −4 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (4, -4). Three points are plotted on the curve at (0, 12), (2, 0) and (6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.

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Graph a quadratic equation in two variables.

  1. Write the quadratic equation with y on one side.
  2. Determine whether the parabola opens upward or downward.
  3. Find the axis of symmetry.
  4. Find the vertex.
  5. Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
  6. Find the x -intercepts.
  7. Graph the parabola.

We were able to find the x -intercepts in the last example by factoring. We find the x -intercepts in the next example by factoring, too.

Graph y = x 2 + 6 x 9 .

Solution

The equation y has on one side. .
Since a is 1 , the parabola opens downward.

To find the axis of symmetry, find x = b 2 a .
. .
.
.
The axis of symmetry is x = 3 . The vertex is on the line x = 3 .
.
Find y when x = 3 . .
.
.
.
The vertex is ( 3 , 0 ) .
.
The y -intercept occurs when x = 0 .
Substitute x = 0 .
Simplify.

The point ( 0 , −9 ) is three units to the left of the line of symmetry.
The point three units to the right of the line of symmetry is ( 6 , −9 ) .
Point symmetric to the y- intercept is ( 6 , −9 )
.
.
.
The y -intercept is ( 0 , −9 ) .
.
The x -intercept occurs when y = 0 . .
Substitute y = 0 . .
Factor the GCF. .
Factor the trinomial. .
Solve for x . .
Connect the points to graph the parabola. .
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Graph the parabola y = −3 x 2 + 12 x 12 .

y : ( 0 , −12 ) ; x : ( 2 , 0 ) ;
axis: x = 2 ; vertex: ( 2 , 0 ) ;
The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -1 to 10. The vertex is at the point (2, 0). One other point is plotted on the curve at (0, -12). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.

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Graph the parabola y = 25 x 2 + 10 x + 1 .

y : ( 0 , 1 ) ; x : ( 1 5 , 0 ) ;
axis: x = 1 5 ; vertex: ( 1 5 , 0 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (-1 fifth, 0). One other point is plotted on the curve at (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1 fifth.

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For the graph of y = x 2 + 6 x 9 , the vertex and the x -intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation 0 = x 2 + 6 x 9 is 0, so there is only one solution. That means there is only one x -intercept, and it is the vertex of the parabola.

How many x -intercepts would you expect to see on the graph of y = x 2 + 4 x + 5 ?

Graph y = x 2 + 4 x + 5 .

Solution

The equation has y on one side. .
Since a is 1, the parabola opens upward. .
To find the axis of symmetry, find x = b 2 a . .
.
.
The axis of symmetry is x = −2 .
.
The vertex is on the line x = −2 .
Find y when x = −2 . .
.
.
.
The vertex is ( −2 , 1 ) .
.
The y -intercept occurs when x = 0 .
Substitute x = 0 .
Simplify.
The point ( 0 , 5 ) is two units to the right of the line of symmetry.
The point two units to the left of the line of symmetry is ( −4 , 5 ) .
.
.
.
The y -intercept is ( 0 , 5 ) .
.
Point symmetric to the y- intercept is ( −4 , 5 ) .
The x - intercept occurs when y = 0 .
Substitute y = 0 .
Test the discriminant.
.
.
b 2 4 a c
4 2 4 1 5
16 20
−4
Since the value of the discriminant is negative, there is no solution and so no x- intercept.
Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.
.
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Graph the parabola y = 2 x 2 6 x + 5 .

y : ( 0 , 5 ) ; x : none ;
axis: x = 3 2 ; vertex: ( 3 2 , 1 2 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (3 halves, 1 half). One other point is plotted on the curve at (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 halves.

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Graph the parabola y = −2 x 2 1 .

y : ( 0 , −1 ) ; x : none ;
axis: x = 0 ; vertex: ( 0 , −1 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (0, -1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 0.

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Questions & Answers

what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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AMJAD
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
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June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
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Ashley
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Ashley
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4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
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operation of trinomial
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y=2×+9
Jacob Reply
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
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Mayra has $124 in her checking account. She writes a check for $152. What is the New Balance in her checking account?
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Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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