An expression is shown below: f(x) = 5x2 + 2x − 3 part a: what are the x-intercepts of the graph of f(x)? show your work. (2 points) part b: is the vertex of the graph of f(x) going to be a maximum or minimum? what are the coordinates of the vertex? justify your answers and show your work. (3 points) part c: what are the steps you would use to graph f(x)? justify that you can use the answers obtained in part a and part b to draw the graph. (5 points) (10 points)

The x-intercepts of a graph are found by factoring.  Set y = 0 because the x-intercepts exist when y = 0.  Put this into the quadratic formula to factor.  Like this:   which simplifies to .  The square root of 64 is 8, so the values we find for x are -1 or 3/5.  Because this is a positive parabola, it opens upwards, like a cup, and a cup has a bottom.  So the vertex is a minimum value.  If it was a negative parabola it would open downwards like a mountain and mountains have top.  The coordinates of the vertex can be found by completing the square.  We also could have factored it by completing the square as well!  Should have read that first...but here's how we do that... Move the constant over to the other side of the equals sign.  The rule for completing the square is that the leading coefficient has to be a 1 and it's a 5.  So we will factor out the 5:  .  Take half the linear term, square it and add it to both sides.  Out linear term is 2/5.  Half of 2/5 is 2/10 or 1/5.  1/5 squared is 1/25.  We will add it into the parenthesis, but don't forget about that 5 hanging out front there, refusing to be ignored.  It is a multiplier, so we multiply 5 by 1/25 and add that to the right.  Like this: .  Doing the math on the right and then rewriting the left using the perfect square binomial we created for the purpose of finding the vertex we have  .  Move the constant back over by subtraction to get .  The vertex is .  Graph it by plotting the x-intercepts and the vertex and the y-intercept.  Find other points and graph them if needed.

A  5x^2  + 2x - 3 = 0
    (5x - 3)(x + 1) = 0

x intercepts are ( 3/5 , 0) and (-1,0)

b   Vertex is a minimum because coefficient of x^2 is positive.
convert to vertex form:-
5(x^2 + 0.4x)  - 3 = 0
5 [ (x + 0.2)^2 - 0.2^2] - 3 = 0
5(x + 0.2)^2 - 0.2 - 3 = 0
5(x + 0.2)^2 - 3.2 = 0
So the coordinates of the vertex are (-0.2, -3.2)

c  Because there is a minimum value t  the graph will be a parabola opening upwards. It will pass through the vertex and the  2 x -intercepts and will be symmetrical about the vertical line x = -0.2

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