a At an x-intercept, we have y=0. At a y-intercept, we have x=0.
B
b The line of symmetry can be found by averaging the x-intercepts.
C
c Use the line of symmetry to find the vertex.
A
a x-intercepts: (- 0.5,0) and (- 1,0)
y-intercept: (0,1) Graph:
B
b x=- 0.75
C
c (- 0.75,- 0.125)
Practice makes perfect
a To draw the function's graph, we need to know a few points through which it passes. Let's make a table of values.
|c|c|c|
[-0.8em]
x & 2x^2+3x+1 & y [0.2em]
[-0.8em]
-3 & 2( -3)^2+3( -3)+1 & 10 [0.2em]
[-0.8em]
-2 & 2( -2)^2+3( -2)+1 & 3 [0.2em]
[-0.8em]
-1 & 2( -1)^2+3( -1)+1 & 0 [0.2em]
[-0.8em]
0 & 2( 0)^2+3( 0)+1 & 1 [0.2em]
[-0.8em]
1 & 2( 1)^2+3( 1)+1 & 6 [0.2em]
[-0.8em]
2 & 2( 2)^2+3( 2)+1 & 15 [0.2em]
From the table of values, we see that the function intersect the y-axis at (0,1). Let's mark the points in a coordinate plane and draw the function's parabola.
A graph intercepts the x-axis when y=0. Therefore, to find the x-intercepts we have to substitute 0 for y and solve the resulting equation using the Quadratic Formula.
We have two x-intercepts, at x=-1 and x=-0.5. Let's summarize the intercepts.
x-intercepts:& (-1,0) and (-0.5,0)
y-intercept:& (0,1)
b All parabolas are symmetric about their vertex. What this means is if two points have the same y-coordinate, such as the x-intercepts, they are equidistant from the parabola's line of symmetry. Therefore, we can find the line of symmetry by averaging the x-intercepts.
The line of symmetry is x=- 0.75.
c From Part A, we know that the line of symmetry is x=-0.75. Since the line of symmetry runs through the parabola's vertex, we can find its y-coordinate by substituting - 0.75 for x in the equation.
