Answer:
down
vertex: (0.25, 4.25)
axis of symmetry : x = 0.25
y-intercept: (0.4)
x-intercept: (-0.781, 0) and ( 1.281, 0)
Explanation:
If we have a quadratic function of the form
[tex]f(x)=ax^2+bx+c[/tex]
then the graph opens downward if a < 1.
The quadratic function we have is
[tex]f(x)=-4x^2+2x+4^{}[/tex]
Since a = -4 < 1, the graph of the function opens downward.
The x-coordinate of the vertex of the function is given by
[tex]x=-\frac{b}{2a}[/tex]
which is also the x-axis of symmetry of the graph ( the axis of symmetry passes through the vertex.
Now in our case a = -4 and b = 2; therefore, the vertex is
[tex]x=\frac{2}{2\cdot4}=\frac{2}{8}[/tex][tex]\boxed{x=0.25.}[/tex]
the above is the x-coordinate of the vertex. The y-coordinate is found by putting x = 0.25 into our function. This gives
[tex]f(0.25)=-4(0.25)^2+2(0.25)+4^{}[/tex]
the above simplifies to give
[tex]f(0.25)=4.25[/tex]
Hence, the coordinates of the vertex are (0.25, 4.25).
The y-intercept is found by putting x = 0 into the function. This gives
[tex]\begin{gathered} f(0)=-4(0)^2+2(0)+4 \\ f(0)=4 \end{gathered}[/tex]
Hence, the y-intercept is at (0,4).
The x-intercepts are the soltuions to
[tex]-4x^2+2x+4=0[/tex]
Using the quadratic formula, the solutions we get are
[tex]x=\frac{-2\pm\sqrt[]{2^2-4(-4)(4)}}{2\cdot(-4)}[/tex]
which simplifies to give
[tex]\begin{gathered} x=-0.781 \\ x=1.281 \end{gathered}[/tex]
Meaning, the x-intercepts are (-0.781, 0) and ( 1.281, 0).
Hence, to summerise our answers
down
vertex: (0.25, 4.25)
axis of symmetry : x = 0.25
y-intercept: (0.4)
x-intercept: (-0.781, 0) and ( 1.281, 0)
The graph of the function is attached below.
