Question

select the correct answer.
what are the factors of this quadratic function?
graph of a parabola opening downward with x-intercepts at (1, 0) and (5, 0)
a. (x - 1) and (x - 5)
b. (x + 1) and (x + 5)
c. (x - 1) and (x + 5)
d. (x + 1) and (x - 5)

Explanation:

Step1: Find x-intercepts

The graph of the quadratic function intersects the x - axis at \(x = 1\) and \(x=5\)? Wait, no, wait. Wait, when \(y = 0\), the roots of the quadratic equation \(ax^{2}+bx + c=0\) are the x - intercepts. If the graph crosses the x - axis at \(x = 1\) and \(x = 5\)? Wait, no, looking at the graph, the parabola crosses the x - axis at \(x = 1\) and \(x = 5\)? Wait, no, let's check again. Wait, the x - intercepts are the points where \(y = 0\). From the graph, the parabola intersects the x - axis at \(x=1\) and \(x = 5\)? Wait, no, when \(x = 1\), \(y = 0\) and when \(x=5\), \(y = 0\)? Wait, no, maybe I made a mistake. Wait, the general form of a quadratic function in factored form is \(y=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots (x - intercepts). If the graph crosses the x - axis at \(x = 1\) and \(x=5\), then the factors would be \((x - 1)\) and \((x - 5)\)? But wait, no, wait the parabola is opening downward. Wait, no, let's check the options. Wait, maybe the roots are \(x=- 1\) and \(x = 5\)? Wait, no, let's look at the graph again. The x - axis is from - 6 to 6. The parabola crosses the x - axis at \(x = 1\) and \(x=5\)? Wait, no, when \(x = 1\), the point is \((1,0)\) and \(x = 5\), \((5,0)\)? Wait, no, maybe I got it wrong. Wait, the factors of a quadratic function \(y = ax^{2}+bx + c\) are of the form \((x - r_1)\) and \((x - r_2)\) where \(r_1\) and \(r_2\) are the roots (x - intercepts). If the graph intersects the x - axis at \(x = 1\) and \(x=5\), then the roots are \(r_1 = 1\) and \(r_2=5\), so the factors are \((x - 1)\) and \((x - 5)\)? But that's option A. Wait, no, wait maybe the roots are \(x=-1\) and \(x = 5\). Wait, no, let's check the options. Option D is \((x + 1)\) and \((x - 5)\), which would mean roots at \(x=-1\) and \(x = 5\). Wait, maybe I misread the x - intercepts. Let's re - examine the graph. The parabola crosses the x - axis at \(x = 1\) and \(x=5\)? No, wait, when \(x = 1\), the point is on the x - axis, and when \(x = 5\), the point is on the x - axis. Wait, but the options: Option D is \((x + 1)\) and \((x - 5)\), which would be roots at \(x=-1\) and \(x = 5\). Wait, maybe the x - intercepts are at \(x=-1\) and \(x = 5\). Let's think again. If the root is \(x=-1\), then the factor is \((x+1)=(x-(-1))\), and if the root is \(x = 5\), the factor is \((x - 5)\). So the factors would be \((x + 1)\) and \((x - 5)\), which is option D. Wait, why? Because when \(x=-1\), \((x + 1)=0\) and when \(x = 5\), \((x - 5)=0\). Let's verify. If the quadratic function has factors \((x + 1)\) and \((x - 5)\), then the roots are \(x=-1\) and \(x = 5\). Looking at the graph, does the parabola cross the x - axis at \(x=-1\) and \(x = 5\)? Let's see, the graph is between x=-6 and x = 6. At x = 1, the graph is above the x - axis? Wait, no, maybe I made a mistake in the x - intercepts. Let's check the y - intercept. The y - intercept is at (0, - 4) or (0, - 5)? Wait, the y - intercept is when \(x = 0\). From the graph, when \(x = 0\), \(y\) is negative. Let's take option D: \((x + 1)(x - 5)=x^{2}-5x+x - 5=x^{2}-4x - 5\). When \(x = 0\), \(y=-5\), which matches the graph (the y - intercept is around - 4 or - 5). Option A: \((x - 1)(x - 5)=x^{2}-6x + 5\), when \(x = 0\), \(y = 5\), which is positive, but the graph has a negative y - intercept, so option A is wrong. Option B: \((x + 1)(x + 5)=x^{2}+6x + 5\), y - intercept is 5, positive, wrong. Option C: \((x - 1)(x + 5)=x^{2}+4x - 5\), y - intercept is - 5, but let's check the roots. Roots at \(x = 1\) and \(x=-5\). The graph does not cross the x - axis at \(x=-5\), so option C is wrong. Option D: \((x + 1)(x - 5)=x^{2}-4x - 5\), roots at \(x=-1\) and \(x = 5\). The graph crosses the x - axis at \(x=-1\) (wait, no, in the graph, the x - intercepts are at \(x = 1\) and \(x = 5\)? Wait, maybe I misread the x - intercepts. Wait, maybe the x - intercepts are at \(x = 1\) and \(x = 5\), but the y - intercept of option A is positive, which doesn't match. Wait, the graph has a negative y - intercept, so the product of the roots (from \(y=ax^{2}+bx + c\), product of roots is \(c/a\)) should be negative (since y - intercept is \(c\), and if \(a\) is negative (parabola opening downward), then \(c/a\) is negative). For option D: \((x + 1)(x - 5)=x^{2}-4x - 5\), if we write it as \(y=-1(x + 1)(x - 5)\) (since it's opening downward), then \(c=-1\times1\times(-5) = 5\)? No, wait, no. Wait, the general form is \(y=a(x - r_1)(x - r_2)\). If the parabola is opening downward, \(a\) is negative. Let's assume \(a=-1\). Then \(y=-1(x + 1)(x - 5)=-1(x^{2}-4x - 5)=-x^{2}+4x + 5\). When \(x = 0\), \(y = 5\), which is positive, but the graph has a negative y - intercept. Wait, I'm confused. Wait, maybe the roots are \(x = 1\) and \(x = 5\), and the quadratic is \(y=- (x - 1)(x - 5)=-x^{2}+6x - 5\). When \(x = 0\), \(y=-5\), which matches the negative y - intercept. Ah! So the factors are \((x - 1)\) and \((x - 5)\), but multiplied by - 1. But the question is about the factors of the quadratic function. The factors (the linear factors) are \((x - 1)\) and \((x - 5)\), which is option A? But wait, when \(x = 0\), \(y=-5\), which matches. Wait, earlier I thought option A has a positive y - intercept, but that's when \(a = 1\). If \(a=-1\), then \(y=- (x - 1)(x - 5)\), which has \(y=-5\) at \(x = 0\), which matches the graph. So the factors are \((x - 1)\) and \((x - 5)\), which is option A. Wait, but let's check the x - intercepts. If the factors are \((x - 1)\) and \((x - 5)\), then the roots are \(x = 1\) and \(x = 5\), which are the x - intercepts. The graph shows x - intercepts at \(x = 1\) and \(x = 5\), and a negative y - intercept, which is consistent with \(y=- (x - 1)(x - 5)\). So the factors are \((x - 1)\) and \((x - 5)\).

Step2: Match with options

The factors of the quadratic function (the linear factors) are \((x - 1)\) and \((x - 5)\), which corresponds to option A.

Answer:

A. \((x - 1)\) and \((x - 5)\)