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Sagot :
For the given function f(x) = 5x² + 2x - 3,
Part A: The x-intercepts of the graph f(x) are: (-1, 0) and (3/5, 0).
Part B: The vertex of the graph of f(X) is going to be a minimum and its coordinates are (-0.2, -3.2).
Part C: The values obtained in the part A and Part B are used for graphing the given function. Since the vertex is minimum, the parabola opens upwards.
How to calculate x-intercepts of a function?
The x-intercepts of a function or equation are obtained at y = 0.
Then, on substituting y = 0 in the function gives the x-coordinate i.e., the x-intercept.
How to calculate the coordinates of the vertex of a function?
To calculate the x coordinate of vertex(h, k), use the formula mentioned below:
h = -b/2a
When the function is in the form of ax² + bx + c.
Then, to calculate the y-coordinate of the vertex, substitute obtained h value in the place of x in the function.
Thus, the required coordinates of the vertex are obtained as (h, k).
Calculation:
It is given that,
f(x) = 5x² + 2x - 3
Part A: Finding the x-intercepts:
Substituting y = 0 i.e., f(x) = 0;
⇒ 5x² + 2x - 3 = 0
On simplifying/factorizing,
⇒ 5x² + 5x - 3x - 3 = 0
⇒ 5x(x + 1) - 3(x +1) = 0
⇒ (5x - 3)(x + 1) =0
∴ x = 3/5 or x = -1
Thus, the x-intercepts are (3/5, 0) and (-1, 0).
Part B: Finding the vertex of the given function:
We have h = -b/2a
From the given function 5x² + 2x - 3; a = 5, b = 2 and c = -3
So, h = -2/2(5) = -1/5 = -0.2
Then, on substituting h = x = -0.2 in the function we get
k = 5(-0.2)² + 2(-0.2) - 3
k = -3.2
Thus, the coordinates of the vertex are (h, k) = (-0.2, -3.2).
Since a > 0 i.e., 5 > 0, the vertex of the graph is going to be minimum and the graph of f(x) opens upwards.
Part C: Steps to graph f(X):
Step 1: Plot the x-intercepts in the coordinate plane
Step 2: Plot the vertex of the function
Step 3: draw a curve line passing through them, opening towards up.
Thus, the graph of the given function is plotted as shown below.
Learn more about the x-intercepts of a function here:
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