Question

the expression $x^2 - 10x + 16$ can be re - written as $(x - 2)(x - 8)$ and $(x - 5)^2 - 9$. which form of the expression best reveals each property? select one expression in each column. | $x$-intercept(s) | $y$-intercept | vertex | line of symmetry | | --- | --- | --- | --- | | $x^2 - 10x + 16$ | $x^2 - 10x + 16$ | $x^2 - 10x + 16$ | $x^2 - 10x + 16$ | | $(x - 2)(x - 8)$ | $(x - 2)(x - 8)$ | $(x - 2)(x - 8)$ | $(x - 2)(x - 8)$ | | $(x - 5)^2 - 9$ | $(x - 5)^2 - 9$ | $(x - 5)^2 - 9$ | $(x - 5)^2 - 9$ |

Explanation:

Step1: Analyze x - intercept(s)

The x - intercepts of a quadratic function \(y = f(x)\) are the values of \(x\) for which \(y = 0\). For a factored form of a quadratic \((x - a)(x - b)\), we can set each factor equal to zero (\(x - a=0\) or \(x - b = 0\)) to find the x - intercepts. For the function \(y=(x - 2)(x - 8)\), setting \(y = 0\) gives \(x-2 = 0\) or \(x - 8=0\), so \(x = 2\) or \(x = 8\). So the factored form \((x - 2)(x - 8)\) best reveals the x - intercepts.

Step2: Analyze y - intercept

The y - intercept of a function \(y=f(x)\) is the value of \(y\) when \(x = 0\). For the standard form \(y=x^{2}-10x + 16\), when \(x = 0\), we substitute \(x = 0\) into the equation: \(y=0^{2}-10\times0 + 16=16\). So the standard form \(x^{2}-10x + 16\) best reveals the y - intercept.

Step3: Analyze Vertex

The vertex form of a quadratic function is \(y=a(x - h)^{2}+k\), where \((h,k)\) is the vertex of the parabola. For the function \(y=(x - 5)^{2}-9\), comparing with \(y=a(x - h)^{2}+k\) (here \(a = 1\), \(h = 5\), \(k=-9\)), the vertex is \((5,-9)\). So the vertex form \((x - 5)^{2}-9\) best reveals the vertex.

Step4: Analyze Line of Symmetry

The line of symmetry of a quadratic function in vertex form \(y=a(x - h)^{2}+k\) is the vertical line \(x = h\). For the function \(y=(x - 5)^{2}-9\), the line of symmetry is \(x = 5\). So the vertex form \((x - 5)^{2}-9\) best reveals the line of symmetry.

Answer:

• x - intercept(s): \((x - 2)(x - 8)\)
• y - intercept: \(x^{2}-10x + 16\)
• Vertex: \((x - 5)^{2}-9\)
• Line of Symmetry: \((x - 5)^{2}-9\)