Exercise 5 Page 255

To draw the graph of the given quadratic equation, which is written in standard form, we must start by identifying the values of a, b, and c. 3x^2-17x+5 = 0 ⇕ y= 3x^2+( - 17)x+ 5 We can see that a= 3, b= - 17, and c= 5. Now, we will follow four steps to graph the equation.

1. Find the axis of symmetry.
2. Calculate the vertex.
3. Identify the y-intercept and its reflection across the axis of symmetry.
4. Connect the points with a parabola.

   Finding the Axis of Symmetry

   The axis of symmetry is a vertical line. Let's recall the equation of the axis of symmetry of a parabola. x=- b2 a Since we already know the values of a and b, we can substitute them into the formula.

   x=- b/2a

   SubstituteII

   a= 3, b= - 17

   x=- - 17/2( 3)

   ▼

   Evaluate right-hand side

   Multiply

   Multiply

   x=- - 17/6

   RemoveNegFracAndNum

   - - a/b= a/b

   x=17/6

   WriteSum

   Write as a sum

   x=12+5/6

   WriteSumFrac

   Write as a sum of fractions

   x=12/6+5/6

   CalcQuot

   Calculate quotient

   x=2+5/6

   AddTerms

   Add terms

   x=2 56

   The axis of symmetry of our parabola is the vertical line with equation x=2 56.

   Finding the Vertex

   To find the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b. Vertex of a Parabola: ( - b/2 a, f ( - b/2 a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which for us is x=2 56. Therefore, the x-coordinate of the vertex is also x=2 56. To find the y-coordinate, we need to substitute 2 56 for x in the given equation.

   y=3x^2-17x+5

   Substitute

   x= 2 56

   y=3( 2 56)^2-17( 2 56)+5

   ▼

   Evaluate right-hand side

   MixedToFrac

   a bc=a* c+b/c

   y=3(17/6)^2-17(17/6)+5

   PowQuot

   (a/b)^m=a^m/b^m

   y=3(17^2/6^2)-17(17/6)+5

   CalcPow

   Calculate power

   y=3(289/36)-17(17/6)+5

   MoveLeftFacToNum

   a*b/c= a* b/c

   y=3* 289/36 -289/6+5

   ReduceFrac

   a/b=.a /3./.b /3.

   y=289/12 -289/6+5

   ExpandFrac

   a/b=a * 2/b * 2

   y=289/12 -578/12+5

   NumberToFrac

   a = 12* a/12

   y=289/12 -578/12+60/12

   AddSubFrac

   Add and subtract fractions

   y=- 229/12

   MoveNegNumToFrac

   Put minus sign in front of fraction

   y = - 229/12

   FracToMixed

   Write fraction as a mixed number

   y=- 19 112

   We found the y-coordinate, and now we know that the vertex is (2 56,- 19 112).

   Identifying the y-intercept and Its Reflection

   The y-intercept of the graph of a quadratic equation written in standard form is given by the value of c. Therefore, the point where our graph intercepts the y-axis is (0, 5). Let's plot this point and its reflection across the axis of symmetry.

   

   Connecting the Points

   We can now draw the graph of the function. Since a= 3, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.

   

   We can't find the exact values of the roots of the equation by looking at the graph. We can, however, see that one of the roots is located between 0 and 1, while the other is located between 5 and 6.