use the parabola tool to graph the quadratic function y=-2x^2+12x-14 graph the parabola by first plotting its vertex and then plotting a second point on the parabola

I used desmos graphing calculator, I was too lazy to do it myself. However, if you wanted to show work, you would have to plug a bunch of (whole) numbers into x to and make a table, then find which output isn't repeated. That input would be the x-coordinate of the vertex, and the output would be the y-coordinate. The second point would be found the same way, using one of the adjacent outputs.

karmenchong karmenchong · Answer 2 · 2024-07-20T23:43:45-04:00

Answer:

See the works below.

Step-by-step explanation:

To graph the quadratic function y = -2x² + 12x - 14, we need to find these points:

determine the direction of the opening and find the vertex
the x-intercept, where the value of y = 0
the y-intercept, where the value of x = 0
(optional) find 2 additional coordinates that reasonably far from the left and right of the above points to make a better graph

Direction of the opening:

If the coefficient of x² < 0 ⇒ the graph opens downwards and the vertex is a maximum
If the coefficient of x² > 0 ⇒ the graph opens upwards and the vertex is a minimum

Since the coefficient of x² is -2, which is smaller than 0, then the graph opens downwards.

Vertex:

[tex]\boxed{(x,y)=\left(\frac{-b}{2a} ,\frac{-(b^2-4ac)}{4a} \right)}[/tex]

[tex]\begin{aligned}(x,y)&=\left(-\frac{12}{2(-2)} ,\frac{-(12^2-4(-2)(-14))}{4(-2)} \right)\\\\&=(3,4)\end{aligned}[/tex]

x-intercept:

When the graph intersect the x-axis, the y-value equals to 0. Therefore, to find the x-intercepts, we substitute y with 0:

[tex]\begin{aligned}y&=-2x^2+12x-14\\0&=-2x^2+12x-14 \end{aligned}[/tex]

Since the equation cannot be factorised, we use the abc formula to find the x-values.

[tex]\boxed{x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} }[/tex]

[tex]\begin{aligned} x&=\frac{-12\pm\sqrt{12^2-4(-2)(-14)} }{2(-2)} \\\\&=\frac{-12\pm4\sqrt{2} }{-4} \\\\&=3\pm\sqrt{2} \\\\&\approx1.59\ or\ 4.41\end{aligned}[/tex]

Hence the x-intercepts are (3+√2, 0) and (3-√2, 0).

Since the x-values are not integer numbers, we pick 2 numbers that are close to the x-intercept.

Let's say we pick x = 1 and 5:

[tex]\begin{aligned}y(x=1)&=-2(1)^2+12(1)-14\\&=(1,-4)\end{aligned}[/tex]

[tex]\begin{aligned}y(x=5)&=-2(5)^2+12(5)-14\\&=(5,-4)\end{aligned}[/tex]

Now we have 2 points (1, -4) and (5, -4).

y-intercept:

When the graph intersect the y-axis, the x-value equals to 0. Therefore, to find the y-intercepts, we substitute x with 0:

[tex]\begin{aligned}y&=-2x^2+12x-14\\&=-2(0)^2+12(0)-14\\&=-14\end{aligned}[/tex]

Hence the y-intercept is (0, -14).

By connecting the above points: (3, 4), (1, -4), (5, -4), and (0, -14), we can graph the function.