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opening upwardaxis of symmetry: x = 0.25vertex: (0.25, -3.25)x-intercepts: 0.25 ±√0.8125 ≈ {-0.651388, 1.151388}y-intercept: -3
Step-by-step explanation:
I find it useful to graph the function using a graphing calculator. That answers most of the questions.
This function is written in standard form (terms have decreasing powers of x). It is a 2nd degree function, because the highest power of x is 2. A second-degree function is also called a quadratic. The leading coefficient is the coefficient of the term with the highest power of x. Here the leading coefficient is 4. The sign of it is positive, which tells you which way the graph opens (when the degree is even).
Opening
When the leading coefficient is positive, the function's graph opens upward. (If it were negative, the graph would open downward.)
Vertex
The vertex is the extreme point on the graph. If you want to find it algebraically, its coordinates are ...
(-b/(2a), c -b^2/(4a))
where 'a', 'b', and 'c' are the coefficients of ax² +bx +c. In this function, a=4, b=-2, c=-3, so the vertex is ...
(-(-2)/(2(4)), -3 -(-2)^2/(4(4))) = (1/4, -3 1/4)
Axis of Symmetry
The axis of symmetry is the vertical line through the vertex. It will have the equation x=constant. Of course, the constant must be the x-value of the vertex (which is why we found the vertex first).
x = 1/4
x-intercepts
The x-intercepts are where the graph crosses the x-axis. The attached graph shows an approximation of them, rounded to thousandths. You can compute the x-intercepts algebraically from the vertex and the leading coefficient.
x-intercepts = (vertex x-value) ± √(-(vertex y-value)/(leading coefficient))
x-intercepts = 1/4 ± √((3 1/4)/(4)) = 0.25 ±√0.8125 ≈ {-0.651388, 1.151388}
y-intercept
The y-intercept is the value of f(0), the point where the graph crosses the y-axis. It is always the constant in the polynomial, named 'c' above. The y-intercept is ...
y-intercept = -3
_____
Additional comment
Sometimes you want the x- and y-intercepts written as coordinates. Of course the y-coordinate of the x-intercepts is 0. Similarly, the x-coordinate of the y-intercept is 0. The coordinate ordered pairs are shown on the graph.
These are fairly standard questions that are asked about quadratic functions. It can be worthwhile to get familiar with the way they are answered. (The computation of the x-intercepts shown here is perhaps a bit unusual, but is very effective once you have the function vertex. The values it gives are correct even when the "x-intercepts" are complex numbers.)
