Answer:
- vertex: (-7/2, -9/4)
- other points: (0, 10), (-5, 0), (-2, 0)
Step-by-step explanation:
The equation 7x +10 = -x² describes two points on a number line: x = -5 and x = -2.
Perhaps you want points to help you plot ...
y = x² +7x +10
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y-intercept
The y-intercept is easy to find: set x=0. What's left is the constant, y = 10. So, the point (0, 10) is one point on the graph.
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vertex
The vertex form of the equation is ...
y = a(x -h)² +k . . . . . . . . where (h, k) is the vertex, and 'a' is the vertical scale factor
If you expand this equation, you get ...
y = a(x² -2hx +h²) +k = ax² -2ahx +ah² +k
Comparing the coefficients to the equation you have, you can see that ...
1 = a
7 = -2ah
10 = ah² +k
Solving for h and k, we find ...
7 = -2(1)h ⇒ h = -7/2
10 = (1)(-7/2)² +k ⇒ k = 10 -49/4 = -9/4
With these values, we can write the vertex form equation as ...
y = (x +7/2)² -9/4
The vertex is (x, y) = (-7/2, -9/4).
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factored form
The equation can be factored by looking for factors of 10 that have a sum of 7. Such values are 2 and 5.
y = x² +7x +10 = (x +2)(x +5)
The x-intercepts are the values of x that make these factors zero.
x +2 = 0 ⇒ x = -2
x +5 = 0 ⇒ x = -5
This tells you two other points on the graph are (-2, 0) and (-5, 0).
